![Page 1: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/1.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Damping and complex modes.
Etienne BalmèsSDToolsEcole Centrale Paris
IMAC 21, Kissimmee
![Page 2: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/2.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Outline
Modes real and complexSpectral decompositionWhen are modes real ?Proportional dampingReal from complex test modesComplex non modes
Damping modelingMaterial, Joint, …Damped FEM modelsLocal/system model
![Page 3: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/3.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Mode ≈ 1 DOF system
![Page 4: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/4.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
1 DOF : influence of damping
![Page 5: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/5.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
1 DOF : frequency domain
1 DOF system (single mode for mechanical system)has 2 complex conjugate poles (linear system modes)
![Page 6: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/6.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
MDOF SISO system
Spectral decompositionMDOF more than 1 pole
SISO Rj is 1x1
MDOF multiple degree of freedomSISO single input single output
![Page 7: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/7.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
MDOF MIMO system
• Sames poles for each IO pair• Residue matrix in spectral decomposition
![Page 8: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/8.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Experimental modes are often “real”
“Complex modes” Residues have a phase spreadPoor modes Have complex residues
“Real modes” Residues for I/O pairs line up
Garteur SM-AG19 test
![Page 9: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/9.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Modes of linear system
• Nominal state-space form
• Left and right eigenvalue problems
![Page 10: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/10.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Modes of linear system
• Mode shape orthogonality and scaling conditions
• Diagonal state-space model
Mode shape Participation factor
![Page 11: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/11.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Modal coordinates, state-space
• Inverting diagonal state-space leads to the spectral decomposition
• Residues of linear systems have no reason to have a single phase (“be real”)
Residue Mode shape Participation factor
![Page 12: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/12.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Normal modes of elastic structure
• Nominal model (elastic + viscous damping)
• Conservative eigenvalue problem
• M>0 & K≥0 ⇒ φ real
![Page 13: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/13.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Normal modes of elastic structure
• Orthogonality• Scaling conditions
• Unit mass• Unit amplitude
• Principal coordinates
![Page 14: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/14.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Modal damping assumption• Assume Γ diagonal
• Leads to second order spectral decomposition
Mode shape Participation factor Residue
![Page 15: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/15.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Modal damping assumptionThis is the only widespread damping modelWhy ?• Compellingly practical• Easy combination of test and analysis• Sufficient mathematical conditions
• Rayleigh• Caughey
Modal also called proportional damping• Often induces small modifications in behaviour
![Page 16: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/16.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Spectral decompositions
• General linear system
• Structure with modal damping
Modal damping ⇔ R is imaginary
![Page 17: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/17.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
When are modes real / complex ?Vibrating structures that are elastic, linear, and
time invariant have real modes.
When are complex modes nearly real ?
Complex modes are found for• Damped structures• Non linear systems• Time varying systems• Periodic structures
Resonances rather than modes
Mathematical trick
Non modal
![Page 18: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/18.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Square beam 1
•One tip damper with variable C•1 pole moves•modes remain nearly real
C
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ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
A case with complex modes
• C=1e3 N/m/s• K1=1.105e5 N/m• K2=1.095e5 N/m• θ=0,20,30,40o
θ• Very close frequencies• Damping and stiffness are not
proportionnal
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ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Losing mode complexity
θDifferent values of frequency
separation ⇒ Modes nearly real
• C=1e3 N/m/s• K1,2=1.1+dk N/m• K2=1.1-dk N/m• θ= 22o
![Page 21: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/21.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
When are complex modes nearly real ?
• Coupling of two modes by viscous damping
Mode 1 responseModal damping
Perturbations fornon-modal damping
![Page 22: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/22.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
When are complex modes nearly real ?
• Uncoupling criterion (Hasselman) ei<<1 ⇔
corresponds to non overlap of peaks• Proof based on damping matrix positiveness
• Generalization uncoupling by block (Balmes 97)
![Page 23: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/23.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Two mode coupling
• Influence on zero & RMS response
• Influence on modeshape complexity
• Effect only significant for high damping
γ12 min, 0, max
![Page 24: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/24.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
First conclusion
Modal damping is a good assumption :• Provided low modal overlap
• Errors on predicted levels are small• Assuming real modes is then OK
![Page 25: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/25.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Experimental modes are often “real”
“Complex modes” Residues have a phase spreadPoor modes Have complex residues
“Real modes” Residues for I/O pairs line up
Garteur SM-AG19 test
![Page 26: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/26.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Real from complex
• Complex modes of second order system verify
• If properness condition verified
![Page 27: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/27.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Real from complex
MIT SERC Active control testbed
28 sensors, 6 independent shaker locations
Balmès (PhD 93, MSSP 97)
![Page 28: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/28.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Real from complex
0.98 -0.02 -0.04 -0.39 0.66 -0.14 0.20 -1.00 -0.87-0.02 1.76 0.13 0.66 0.06 -0.49 -0.85 -1.05 -0.00-0.04 0.13 1.80 -0.06 -0.24 0.13 0.75 1.39 2.13-0.39 0.66 -0.06 4.68 1.10 -1.08 -0.39 -1.78 2.460.66 0.06 -0.24 1.10 6.07 -1.69 0.64 -2.00 -1.41
-0.14 -0.49 0.13 -1.08 -1.69 11.66 -3.14 4.11 0.970.20 -0.85 0.75 -0.39 0.64 -3.14 3.54 -0.61 -0.12
-1.00 -1.05 1.39 -1.78 -2.00 4.11 -0.61 4.56 1.07-0.87 -0.00 2.13 2.46 -1.41 0.97 -0.12 1.07 12.72
• Γ consistent from test to test
• Proportional damping response significantly different
![Page 29: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/29.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Real from complex : but …
• Real data is almost never that clean• Use of Γtest difficult and does not change designThus for modal test derived damping• assume modal damping• if you really want real modes
– Take the imaginary part of the residue– Use appropriation (Foltete 98)– Use transformations (Niedbal 84, Zhang 85, Wei 87,
Imregun 93, Balmes 93 97, Ahmadian 95, …)
![Page 30: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/30.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Complex non modes I
Frequency shifts in batch tests induce complexity
![Page 31: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/31.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Complex non modes II
Simulation : frequency change of 0.2%
Frequency shifts in batch tests induce complexity
![Page 32: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/32.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Computational complex shapes
Propagating waves •boundary elements•cyclic symmetry
![Page 33: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/33.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Complex shapes
• Non modes– Non linear response– Poor identification– Non invariance of test article – Signal processing distortions– Operational deflection shapes, propagating waves– …
• Mathematical trick (cyclic symmetry)• True complex modes (damped linear system)
![Page 34: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/34.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Modeling damping
• Local models (joints and materials)
• Finite element models, damping design tools
• Simplifying assumptions for system dynamics (dynamic behaviour rather than local knowledge models)
![Page 35: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/35.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Joints
F = f(q,t)F = k(s)q(s)
Non linear models : only practical if local
![Page 36: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/36.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Viscoelastic materials
ε
σΕsη
−Εsη
−1 1Elastic
Viscoelastic
![Page 37: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/37.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Viscoelastic constitutive relations• Stress is a function of strain history • Complex modulus in Laplace domain
![Page 38: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/38.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Reduced frequency nomograms
![Page 39: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/39.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Viscoelastic constitutive laws
Viscous Histeretic Standard viscoelastic
Alternatives : • More relaxation constants• Fractional derivatives• Direct use of experimental master curve
K
![Page 40: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/40.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Material damping models
Stress/strain curveSimple models
K+iB
![Page 41: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/41.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Consitutive model order1 pole model (3 parameter) loss factor is wrong
3 pole model : better matchin band but not very good outside
Good models require high order
![Page 42: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/42.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Other sources of dissipation
• Non linearities – Material (plasticity, …), joint– Contact (friction dampers, joint damping, micro-slip,
…)• Coupling with other media
– Radiation in air, water, soil, etc.– Gyroscopic damping– Electrical systems (active control)– Particle filled cavities– Lubrication, …
![Page 43: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/43.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Frequency dependent models• Dynamic stiffness : linear combination of fixed
matrices
• Direct frequency response
• Non-linear eigenvalue extraction
![Page 44: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/44.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Frequency independent models
• Material formulation with internal fields (rational and fractional derivates)
Trick increase model orderto gain frequency idependence
![Page 45: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/45.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model • Proved methodologies ADF (Lesieutre),
GHM (Gola, …) , Prony series (abaqus)
+ Integrates into standard solvers
+ Time equivalent
– High order for good material (solvers need to account for block structure)
Frequency independent models
![Page 46: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/46.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
A major computational challenge
• Sandwich oil pan• 58766 DOFs
![Page 47: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/47.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
A major computational challenge
• NASTRAN 70.7 & SDT 5• 58766 DOFs
10 temperatures 1000 frequency
NASTRAN direct : 9 daysNASTRAN direct : 9 daysNASTRAN direct : 9 daysNASTRAN direct : 9 daysSDT Iterative : 612 sSDT Iterative : 612 sSDT Iterative : 612 sSDT Iterative : 612 sSpeedup : 1300Speedup : 1300Speedup : 1300Speedup : 1300 Still work to be done on
accuracy/performance trade-off
More examples here
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ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
System vs. local
• System models (dynamic behaviour)
• FEM models (geometry/material knowlege)
• Each objective requires different assumptions
![Page 49: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/49.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
System damping. 1 DOF example.a) Bodeb) Nyquistc) 1DOF modeld) Dynamic stiffness
dynamic stiffness equal at resonance ⇒
Response is the same
Different local models, same system response
![Page 50: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/50.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Proportional damping model• Modal damping is a good for system not for local• Forced response along first mode
True modal
Modal looses localization of damping
![Page 51: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/51.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Systems and material level tests• Based on system level test, you get
test derived damping ratio for system model they– are difficult/impossible to extrapolate to other system
configurations– require matching of test/FEM modes– can rarely be translated in local damping information
• Based on materials/components tests, you get– damped FEM models, which are still difficult to solve– are only valid if almost all damping comes from well
characterized parts• Most damping models are arbitrary design parameters
![Page 52: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/52.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Conclusion
• A lot of complex shapes are just not modes• Some of them do come from damping• Damping is becoming part of design• We will see more of complex modes
www.sdtools.com/Publications.html
![Page 53: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/53.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
System damping model
Balmès IMAC 97 : you can build equivalent system models in modal coordinates
![Page 54: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/54.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
1 % damping : viscous & structural
To obtain 1% modal damping• structural material damping leads to sparse matrix• Viscous material damping leads to full matrix
![Page 55: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/55.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Complex modes history
Just a few authors who talked about complex modes at IMACBalmes 94, Chung 87, Debao 87, Ewins 93, Gladwell 95, Hamidi 89, Ibrahim 83, 93 Imregun91, 93, Inman 86, 95, Jun 84, Kirshenboin 87, Lallement 84, 87, Mitchell 90, 92, Montgomery 93, Niedbal 84, Ozguven 82, 86, Sas 92, Sestieri93, Wei 87, Wicks 96, Zhang 84 …
![Page 56: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/56.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Floor panel (7998 nodes, 7813 elements) + free layers 2195 nodes, 1908 elements or+ constrained layer 6595 n, 3816 elts)
NASTRAN element formulations
Floor pannel design
Objective :Objective :Objective :Objective : propose design stepspropose design stepspropose design stepspropose design stepsValidation Validation Validation Validation addressed elsewhereaddressed elsewhereaddressed elsewhereaddressed elsewhere
![Page 57: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/57.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Design problems
• Basic design : selection of relevant materials• Thickness optimization• Treatment nature and position
– A1 free layer 2.47 mm viscoelastic– B1 constrained layer : 50 mm visco, 0.3 mm steel
![Page 58: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/58.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Material Selection
1. Select frequency range2. Select temp range3. Validate relevance on
nomogram
Other considerationsManufacturing, price,
outgazing, aging, oil,…
SM50e
PL3023
![Page 59: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/59.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Temperature robustness validation
For a selected design performance is judged byFRFs and Poles
Sensitivity to temperature must be evaluated B1/Ta
B1/SM50e
![Page 60: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/60.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Thickness optimization
![Page 61: Damping and complex modes - sdtools.com · 2 complex conjugate poles (linear system modes) Modes Spectral decomposition When real ? Proportional damping Real from complex Complex](https://reader031.vdocument.in/reader031/viewer/2022020122/5d5af72088c993eb1c8b94f2/html5/thumbnails/61.jpg)
ModesSpectral decompositionWhen real ?Proportional dampingReal from complexComplex non modes
DampingMaterial, Joint, …Damped modelsLocal/system model
Design comparisons
• A1(Free layer) mass = 2 x B1 mass
• B1-Ta very as efficient as A1 at 20oC
• B1-SM very robust but not very good