Introduction to Iwan Models and Modal Testing for Structures with Joints

Transcription

1 Introduction to Iwan Models and Modal Testing for Structures with Joints Matthew S. Allen Associate Professor University of Wisconsin-Madison Thanks to Brandon Deaner (now at Mercury Marine) for his work in this area and for creating the initial version of these slides. Motivation Much of the damping in built up structures comes from the interfaces (joints). Joints are nonlinear and influenced by physics at multiple length scales. However, the response of structures with joints is often quasilinear. m mm. mm Assembly / Structural scale FEA + joint models FEA, detailed joint mechanics m Meso, micro, asperity models Magnitude size Frequency (Hz) nm Molecular dynamics. nm Frequency (Hz) Quantum mechanics, chemistry

2 Outline u u 2 u 3 K Discrete Damping Model Modal Damping Model Apply the Modal Damping Model Academic System Finite Element Model Beam Structure Exhaust Structure Conclusions K M K M F S, K T, χ, β M F We want to accurately model the nonlinear damping seen in joints. Structures with joints exhibit increased damping. Jointed Structure vs. Monolithic Structure Response Response Time Time Damping depends on the amplitude of the force 2

3 The nonlinear damping of joints is classified into two regions. nonlinearity associated with the slip region at the outskirts of the contact patch. Stick Region F macro-slip occurs when the stick region vanishes and there is large relative motion between the two parts. Slip Region F D. J. Segalman, "An Initial Overview of Iwan Modeling for Mechanical Joints," Sandia National Laboratories, Albuquerque, New Mexico SAND2-8, 2. A Jenkins element is often used to provide stick slip behavior. x(t, ϕ) F k Consists of: Coulomb Friction Damper Linear Elastic Spring Provides Stick-Slip behavior Stuck state has stiffness and no energy dissipation Slipped state has no stiffness and energy dissipation 3

4 One approach is to model a joint with several Jenkins elements over an area. * C. W. Schwingshackl, D. D. Maio, I. Sever, and J. S. Green, "Modeling and validation of the nonlinear dynamic behavior of bolted flange joints," Journal of Engineering for Gas Turbines and Power, vol. 35, 23. (see also) Bograd, S., Reuss, P., Schmidt, A., Gaul, L., and Mayer, M., Modeling the Dynamics of Mechanical Joints, Mechanical Systems and Signal Processing, 25, pp , 2. Advantage: Predictive! Disadvantage: Many degrees-of-freedom, no guarantee that power law behavior is captured. A parallel arrangement of Jenkins elements is known as an Iwan model and is used to model both regions of slip. Iwan Model x(t, ϕ ) Micro-Slip Macro-Slip x(t, ϕ 2 ) F x(t, ϕ 3 ) x(t, ϕ 4 ) D. J. Segalman, "A Four-Parameter Iwan Model for Lap-Type Joints," Journal of Applied Mechanics, vol. 72, pp , September 25. 4

5 A 4 Parameter Iwan model is used to capture both micro and macro slip. The Iwan Model consists of spring and frictional damper elements in parallel x(t, ϕ ) Population Density x(t, ϕ 2 ) ϕ max F x(t, ϕ 3 ) ρ(ϕ) x(t, ϕ 4 ) macro-slip ρ(ϕ) = R ϕ χ [H(ϕ) H(ϕ ϕ max )]+Sδ(ϕ ϕ max ) D. J. Segalman, "A Four-Parameter Iwan Model for Lap-Type Joints," Journal of Applied Mechanics, vol. 72, pp , September 25. ϕ We can represent the 4 Parameter Iwan model graphically to describes how stiffness and dissipation change with force level. Converted to physical parameters {R, χ, ϕ max, S} {F S, K T, χ, β} F S F S Slope = Stiffness K T macro-slip log(dissipation/cycle) Slope = 3 + χ β macro-slip Joint Force log(joint Force) This is the key to the Iwan model: it captures a specific evolution of damping and stiffness with amplitude that has been observed in many experiments!! 5

6 Stiffness changes are typically small damping effect is of primary interest. mx cx kx F t E m X E diss / cyc c X cos( ) 2 m n E diss/cyc goes as X 2!! X 2 F S diss / cyc / d n 2 2 Vt () Vt ( ) d 2 d n X Log Damping Ratio no-slip (material damping) Slope = + χ macro-slip Slope = Log Joint Force Why would we want to apply the damping in a modal framework? Simplify computations for structure with many joints Many parameters needed to characterize each joint separately. F,K,χ S T, β F S,K T,χ, β F S,K T,χ, β 6

7 Why would we want to apply the damping in a modal framework? Simplify computations for structure with many joints K M = F S, K T, χ, β q 2 T q, Φ F X ˆ r rq r r +Fr J Why would we want to apply the damping in a modal framework? Because tests in the regime frequently reveal that the structure as a whole often behaves modally! 7

8 The proposed modal Iwan model comes with several assumptions. Linear Modes of the system are preserved No geometric nonlinearities due to large deflection or material nonlinearities F No coupling among modes r th applied = r th Modal Response only D Does the modal approach work for a simple mass spring system? Discrete Model: u u 2 u 3 K K K F M M F S, K T, χ, β M Modal Models:??? K K 2 K 3 M = M = M = F S, K T, χ, β F S2, K T2, χ 2, β 2 F S3, K T3, χ 3, β 3 q q 2 q 3 D. J. Segalman, "A Modal Approach to Modeling Spatially Distributed Vibration Energy Dissipation," Sandia National Laboratories, Albuquerque, New Mexico and Livermore, California SAND2-4763, 2. 8

9 Comparison procedure for discrete and modal simulations. Discrete Simulation: F S, K T, χ, β Apply Impact Force Force Time (s) Convert physical response to modal q = - x (a) Natural Frequency (Hz) Hilbert Transform Modal Joint F (b) Modal Energy Dissipation per cycle Modal Joint F Modal Simulation: F S, K T, χ, β Apply Impact Force Force Time (s) Optimize Analytical Model to frequency and energy dissipation to each mode. Min f (a) Natural Frequency (Hz) Hilbert Transform 24 2 Modal Joint F (b) Modal Energy Dissipation per cycle Modal Joint F Natural Frequency (Hz) Compare Modal and Discrete simulations Modal Energy Dissipation/Cycle (a) Natural Frequency (Hz) Hilbert Transform 24 2 Modal Joint F (b) Modal Energy Dissipation per cycle Modal Joint F Hilbert Transform with polynomial fitting is used to extract frequency and energy dissipation data. Academic Model 3rd Mode Velocity Angle (rad) Time (s) Measurement Polynomial Fit Measurement Hilbert Transform Polynomial Fit Instantaneous damping and frequency are related to the derivative of the amplitude and phase, respectively Time (s) 9

10 Model T ˆ J J S S Optimize analytical model to match frequency and energy dissipation to each mode. Optimization can be used to fit the analytical model to measured changes in dissipation and frequency. Natural Frequency macro-slip log(modal Dissipation/Cycle) macro-slip Modal Joint Force ˆ ˆ rˆ K ˆ T K ( ˆ D )( ˆ 2) fˆ f Model 2 max Kˆ 2 Exp Model Exp Model ˆ ˆ 3 4Rq if ˆ J ˆ ˆ J if F Fˆ Approximate S ˆ F F S D ˆ 3 ˆ 2 Dˆ ˆ Model ˆ ˆ Exp DModel Dissipation: max fexp f 4 ˆ Model ˆ J ˆ J ˆ qf ˆ if F F S if F FS S log(modal Joint Force) Optimize the Analytical Modal Model in a least 2 squares sense Dˆ Dˆ fˆ fˆ Min Approximate f f f Frequency: 2 How well does a modal Iwan model predict the response of a spring mass system. vs. fˆ ˆ rˆ K Kˆ ( ˆ )( ˆ 2) 2 Kˆ 2 ˆ ˆ 3 4Rq ˆ J if ˆ ˆ F F D ˆ 3 ˆ 2 Model 4 ˆ ˆ J qf ˆ S if F F if if Fˆ Fˆ Fˆ Fˆ S S vs. Natural Frequency (Hz) Academic Model st Mode 2 3 macro-slip Discrete Sim Analytical Model Modal Sim Modal Energy Dissipation/Cycle Academic Model st Mode 5 Discrete Sim Analytical Model 4 Modal Sim macro-slip 2 3

11 How well does a modal Iwan model predict the response of a spring mass system. Natural Frequency (Hz) Academic Model 2nd Mode 2 3 Discrete Sim Analytical Model Modal Sim Modal Energy Dissipation/Cycle Academic Model 2nd Mode 2 3 Discrete Sim Analytical Model Modal Sim Natural Frequency (Hz) Academic Model 3rd Mode 2 3 Discrete Sim Analytical Model Modal Sim Modal Energy Dissipation/Cycle Academic Model 3rd Mode 2 3 Discrete Sim Analytical Model Modal Sim Limitations The previous results all excited the system with a force that was spatially distributed to excite a single linear mode. Forces with other spatial distribution can cause macro-slip at a lower force level. Natural Frequency (rad) Frequency vs Log Modal Amplitude Log Modal Disp. Amplitude Modal Energy Damping Dissipation/Cycle Ratio Log Damping Log Dissipation Ratio vs Log vs. Modal Amplitude Linear Material region Damping governed by Modal Disp. Disp. Amplitude Amplitude

12 Limitations Mode 2 Frequency vs Log Modal Amplitude Damping Ratio vs Log Modal Amp. Natural Frequency (rad) Log Modal Disp. Amplitude Damping Ratio Modal Disp. Amplitude Natural Frequency (rad) Limitations Mode 3 Frequency vs Log Modal Amplitude Damping Ratio vs Modal Amplitude Log Modal Disp. Amplitude Damping Ratio Modal Disp. Amplitude 2

13 Can an Iwan model be used to predict the response of a FE model? Academic simulations: region = Excellent Representation transition region = may be inaccurate macro-slip region = Excellent Representation Finite Element models? Comparison procedure for discrete and modal simulations. Discrete Simulation: Apply Impact Force Convert physical response to modal q = * - x (a) Natural Frequency (Hz) Hilbert Transform Modal Joint F (b) Modal Energy Dissipation per cycle Modal Joint F Apply Impact Force Modal Simulation: Optimize Analytical Model to frequency and energy dissipation to each mode. Min f (a) Natural Frequency (Hz) Hilbert Transform 24 2 Modal Joint F (b) Modal Energy Dissipation per cycle Modal Joint F Natural Frequency (Hz) Compare Modal and Discrete simulations Modal Energy Dissipation/Cycle

14 How is a discrete Iwan joint applied in a finite element model? FE model Determine the contact surfaces Use an Iwan model for each joint z y x Contact Region Discrete Iwan (F s, K T, χ, β) MSA3 A simple FE model was created for simulations. Mesh: Hexahedron elements Relatively course mesh (,554 nodes) 2" 3.5" Constraints: Free-Free Boundary conditions Two washers are merged with the link Washers are constrained to move and rotate along the surface of the beam Link 2" Washers Beam Discrete Iwan Models 4

15 Slide 28 MSA3 "For Simulations" isn't too specific. How to improve? "To evaluate the modal approach"?? Did you sufficiently explain the motivation - Why did we create an FEA model? What are we hoping to learn from these simulations? Matt Allen, /3/23

16 How well does the modal Iwan model predict the response of the structure? Natural Frequency (Hz) lbf Finite Element Model st Mode 5 lbf lbf 5 lbf lbf Discrete Sim Analytical Model Modal Sim macro-slip 5 lbf lbf lbf 5 lbf Modal Energy Dissipation/Cycle lbf -6 5 lbf Finite Element Model st Mode Discrete Sim Analytical Model Modal Sim = -.28 lbf 5 lbf lbf lbf 5 lbf macro-slip lbf 5 lbf How well does the modal Iwan model predict the response of the structure? Mode 2 Natural Frequency (Hz) lbf Finite Element Model 2nd Mode 5 lbf lbf 5 lbf lbf 5 lbf Discrete Sim Analytical Model Modal Sim lbf 5 lbf lbf Modal Energy Dissipation/Cycle lbf Finite Element Model 2nd Mode Discrete Sim Analytical Model Modal Sim = lbf lbf 5 lbf lbf 5 lbf lbf 5 lbf lbf

17 How well does the modal Iwan model predict the response of the structure? Mode 2 Natural Frequency (Hz) lbf Finite Element Model 2nd Mode 5 lbf lbf 5 lbf lbf 5 lbf Discrete Sim Analytical Model Modal Sim lbf 5 lbf lbf Modal Energy Dissipation/Cycle lbf Finite Element Model 2nd Mode Discrete Sim Analytical Model Modal Sim = lbf lbf 5 lbf lbf 5 lbf lbf 5 lbf lbf How well does the modal Iwan model predict the response of the structure? Mode 3 Natural Frequency (Hz) lbf Finite Element Model 3rd Mode 5 lbf lbf lbf 5 lbf 5 lbf lbf Discrete Sim Analytical Model Modal Sim macro-slip lbf 5 lbf Modal Energy Dissipation/Cycle lbf Finite Element Model 3rd Mode Discrete Sim Analytical Model Modal Sim 5 lbf lbf lbf 5 lbf 5 lbf lbf 5 lbf macro-slip lbf 6

18 The response can be simulated with a few SDOF modal simulations and compare well in the micro slip region. Modal Velocity Modal Velocity Modal Velocity 5 st Mode Discrete Sim Modal Sim Time (s) 2nd Mode Time (s) 3rd Mode 5 Modal Responses Time (s) The response can be simulated with a few SDOF modal simulations and compare well in the micro slip region. Physical Responses FE Model with 5lbf Impact Force 25 Discrete Sim Modal Sim Z-Velocity (m/s) at Midpoint Time (s) 7

19 Can a modal Iwan model be used to predict the response of actual experimental data. FE simulations: region = Good Representation macro-slip region = Fair Representation Laboratory data? Laboratory tests were initially conducted on a beam with a link attached in the middle. 3.5" Same dimensions as the FE model 2" Washers between link and beam were removed Bolt Link Beam 2" Washer Nut The damping of this structure was found to be very small (joint carries little load) and hence a modified structure was studied instead. 8

20 Alternative: Two beam test structure Beam Suspension 2 strings 8 elastic bungees Forcing Automatic hammer Consistent force input The two beam test structure shows measurable, nonlinear damping. Damping (%) at various torque levels: in-lbs, 3 in-lbs, 5 in-lbs Elastic Mode # in-lbs Torque, ζ (%) Percent Difference (%) 3 in-lbs Torque, ζ (%) Percent Difference (%) 5 in-lbs Torque, ζ (%) Percent Difference (%)

21 Filtered Average Time (s) Measurements were taken with a scanning laser Doppler vibrometer. Velocity Measurements: Polytec OFV-534 Single Point LDV used as reference. Polytec PSV-4 Scanning Head used to measure 7 points on the structure. Comparison procedure for discrete and modal simulations. Laboratory Measurements: Apply Impact Force Force Time (s) Convert physical response to modal Modal Velocity (a) Natural Frequency (Hz) Hilbert Transform Modal Joint F (b) Modal Energy Dissipation per cycle Modal Joint F Apply Impact Force Modal Simulation: Optimize Analytical Model to frequency and energy dissipation to each mode. Min f Hilbert Transform Natural Frequency (Hz) Compare Modal and Discrete simulations Modal Energy Dissipation/Cycle Force Time (s) (a) Natural Frequency (Hz) Modal Joint F (b) Modal Energy Dissipation per cycle Modal Joint F 2

22 Filtered Average Time (s) Apply Impact Force Force Time (s) Compare the standard deviation of the trimmed set versus untrimmed set. Torque (in-lbs) Hammer Level ( lowest - 4 highest) Mean Impact Force (N) Standard Deviation of Impact Force (N) Standard Deviation of Trimmed (N) Modal Hammer Modal Hammer Convert physical response to modal Modal Velocity Next the measurements were To divided isolate by the the modes, mass measurements normalized were mode band shape pass value. filtered. st Elastic Mode 2.9 Hz 2nd Elastic Mode 24.8 Hz All Measurements Filtered Filtered Average Modal Velocity. -. Velocity (m/s) rd Elastic Mode Hz th Elastic Mode Hz th Elastic Mode 83.2 Hz 6th Elastic Mode 5 Hz Frequency (Hz) Time (s) -. 2

23 Optimize Analytical Model to frequency and energy dissipation to each mode. Min f The modal Iwan model seems to have difficulty capturing the energy dissipation and frequency. Two Beam st Mode 3 in-lbs Two Beam st Mode 3 in-lbs Natural Frequency (Hz) Lab Data 3 Modal Iwan Model -2 - Modal Energy Dissipation/Cycle Lab Data Modal Iwan Model How do we account for the damping associated with the material and boundary conditions? Add a viscous damper to the modal Iwan model K F S, K T, χ, β M = C q 22

24 The modal Iwan model with a viscous damper captures the laboratory data accurately! Natural Frequency (Hz) Two Beam st Mode 3 in-lbs 5 4 Lab Data Modal Iwan Model 3 Modal Iwan & VD Model -2 - Modal Energy Dissipation/Cycle Two Beam st Mode 3 in-lbs Lab Data Modal Iwan Model Modal Iwan & VD Model VD Model -2 - The response can then be simulated as a superposition of nonlinear SDOF systems..5 Filtered Data Simulated Model. Velocity (m/s) at Midpoint Time (s) 23

25 These concepts also hold for realistic hardware Front and rear catalytic converters assembled together with required assembly torque and exhaust manifold gasket. System hung freely suspended by bungee cords to complete a roving hammer test. Composite FRF [m/s 2 ] Linear Experimental Modal Analysis Composite FRF for Coupled Catalyic Converter System Combined Composite FRF Composite FRF X Composite FRF Z Force levels kept low to estimate linear damping Natural Modal Damping Frequency Index Ratio 5 [Hz] Frequency [Hz] Deflection Type Bending Bending Localized Heat Shield Mode Localized Heat Shield Mode Torsion Bending 24

26 Damping Ratio Modal Iwan model accurately captures the damping versus amplitude for various input points (various combinations of the different modal amplitudes)! Damping vs. Velocity Amplitude Strike 24 z Strike 34 z Strike 34 x Strike 24 z2 Strike 34 z2 Strike 34 x2 Because the quasi-modal framework is valid, we can reduce the number of tests and test/model comparisons dramatically! Strike 24 z3 Strike 34 z3-2.6 Strike 34 x Amplitude (m/s) Preview: Modal Iwan parameters can(*) be used to deduce discrete joint parameters. Measurements of modal response in regime used to deduce discrete joint properties. [Segalman, Allen, Eriten & Hoppman, ASME-IDETC 25] 25

27 Conclusions Modal Iwan models can represent real structures quite accurately! Academic Model region = Excellent Representation macro-slip region = Excellent Representation FE Model region = Good Representation macro-slip region = Fair Representation Laboratory Data region = Excellent Representation macro-slip region = Not Obtained Acknowledgements Some of the work described was conducted at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC4-94AL85. Special thanks to Simulation Work: Michael Starr, Daniel Segalman, Michael Guthrie, Brandon Deaner, Robert Lacayo Lab Work: Jill Blecke, Matthew Allen, Hartono Sumali, Michael Starr, Randy Mayes, Brandon Zwink, Patrick Hunter, Dan Segalman, Steve Myatt Univ. Wisconsin Research Group: Former: Brandon Deaner, Robert Kuether, Hamid Ardeh, Dan Roettgen, David Ehrhardt, Current: Robert Lacayo, Kurt Hoppman, Melih Eriten, B. Deaner, M. S. Allen, M. J. Starr, D. J. Segalman, and H. Sumali, "Application of Viscous and Iwan Modal Damping Models to Experimental Measurements From Bolted Structures," ASME Journal of Vibrations and Acoustics, vol. 37, p. 2, 25. B. J. Deaner, M. S. Allen, M. J. Starr, and D. J. Segalman, Application of Viscous and Iwan Modal Damping Models to Experimental Measurements from Bolted Structures," presented at the International Design Engineering Technical Conferences, Portland, Oregon USA, 23. M. S. Allen, R. J. Kuether, B. Deaner, and M. W. Sracic, "A Numerical Continuation Method to Compute Nonlinear Normal Modes Using Modal Reduction," presented at the 53rd Structures, Structural Dynamics, and Materials Conference (SDM), Honolulu, Hawaii,

28 Response Excitation Alternative T/F Analysis: Zeroed Early time FFT Response Zero Pts Time (ms) High Amplitude Nonlinear Low Amplitude Linear Nonlinearity is assumed to be active at high amplitudes and inactive at lower amplitudes The response then becomes more linear as more of the initial nonlinear response is nullified. Impulse responses with initial segments of varying length set to zero are compared in the frequency domain. The nonzero portion of each impulse response begins at a point in which the response is near zero. Magnitude Response Excitation Alternative T/F Analysis: Zeroed Early time FFT NLDetect: FFT of Time Response - Truncated at zero points) Response Zero Pts Time (ms) High Amplitude Nonlinear Low Amplitude Linear Frequency (Hz) Nonlinearity is assumed to be active at high amplitudes and inactive at lower amplitudes The response then becomes more linear as more of the initial nonlinear response is nullified. Impulse responses with initial segments of varying length set to zero are compared in the frequency domain. The nonzero portion of each impulse response begins at a point in which the response is near zero. 27

29 Beam: ZEFFTs Magnitude NLDetect: FFT of Time Response - Truncated at zero points) and 275 Hz modes clearly soften as the amplitude increases. FEA Mode : 644 Hz 4 FEA Mode 3: 58 Hz Frequency (Hz) FFT of Time Response - Truncated points) NLDetect: at zero All of these lines come from one response with different parts erased! Magnitude FEA Mode 6: 335 Hz Frequency (Hz) Magnitude 7 Beam: BEND and IBEND NLDetect: FFT of Time Response - Truncated at zero points) 4 Fit to 4 3 Extrap from 4 to Extrap from 4 to Extrap from 4 to Extrap from 4 to IBEND: Somewhat nonlinear before 7ms, Significantly nonlinear before 5ms Frequency (Hz).4 Normalized Integral Normalized Integral of Difference FRF over Frequency Integral fit time time (ms)

30 Linear Beam Magnitude NLDetect: FFT of Time Response - Truncated at zero points) Results on previous slides for 2 lbf peak force This shows the ZEFFTs for a 4 lbf peak force 3 Hz shift observed from high to low amplitude Frequency (Hz) 29