Can the Spatial Distribution of Damping be Measured? S. A DHIKARI , J. W OODHOUSE AND A. S RIKANTH P HANI Cambridge University Engineering Department Cambridge, U.K. Can the Spatial Distribution of Damping be Measured? – p.1/22
Outline of the Talk Introduction Models of damping: Viscous and non-viscous damping Complex frequencies and modes Theory of damping identification Numerical Results Experimental Results Conclusions Can the Spatial Distribution of Damping be Measured? – p.2/22
✢ ✦ ✗ ✜ ✛ ✍ ✏ ✜ ✍ ✞ ✔ ✝ ✩ ✣ ✍ ✧ ✤ ★ ✒ ✍ ✕ ✫ ✍ ✄ ✒ ✫ ✍ ✍ ✖ ✒ ✠ ✭ ✜ ✩ ✝ ✞ ✮ ✟ ✁ ✜ ✝ ✞ ✁ ✕ ✕ ✝ ✠ ✩ ✍ ✒ ☛ ✏ ✩ ✍ ✞ ✔ ✬ Viscously Damped Systems Equations of motion: ✄✆☎ ✄✆☎ ✄✆☎ M �✂✁ C K (1) ✡☞☛ Approximate Complex frequency and modes: ✖✪✩ ✌✎✍ ✑✓✒ ✘✚✙ ✣✥✤ : Undamped natural frequency, x : Undamped modes ✖✪✩✭ C are the elements of the damping matrix in modal coordinates. Can the Spatial Distribution of Damping be Measured? – p.3/22
Some General Questions of Interest 1. From experimentally determined complex modes can one identify the underlying damping mechanism ? Is it viscous or non-viscous? Can the correct model parameters be found experimentally? 2. Is it possible to establish experimentally the spatial distribution of damping? 3. Is it possible that more than one damping model with corresponding correct sets of parameters may represent the system response equally well, so that the identified model becomes non-unique ? 4. Does the selection of damping model matter from an engineering point of view? Which aspects of behaviour are wrongly predicted by an incorrect damping model? Can the Spatial Distribution of Damping be Measured? – p.4/22
✍ ✄ ✖ ✡ ✍ ✒ ★ ✤ ✧ ✣ ✦ ✢ ✍ ✔ ✞ ✒ ✜ ✏ ✍ ✛ ✍ ✗ ✝ ✩ ✄ ✝ ✜ ✖ ✄ ✩✭ ✝ ✒ ✄ ☛ ☛ ✩ ✝ ✒ ✩ ✫ ✒ ✬ ✍ ✫ ✒ ✄ ✝ ✍ ✍ ✄ ✡ ✬ ☎ ✆ ✝ ☎ ✄ ✁ ✟ ✝ ☎ ✝ ✝ ✄ ✂ ✁ � ✞ ✝ ✜ ✁ � ✭ ✞ ✁ ✍ ✜ ✍ ✖ ✡ ✔ ✞ ✍ ✠ ✏ ✍ ✌ ✮ ✍ ✞ ✩ ☛ ✝ ✄ ✄ ✒ ✡ ✠ ✝ ✝ Non-viscously Damped Systems Equations of motion: ✄✆☎ ✄✆☎ ✄✆☎ M (2) ✄✆☎ is ✞✠✟ matrix of kernel functions. Approximate Complex frequency and modes: ✑✓✒ ✑✓✒ ✘✚✙ ✣✥✤ ✄✆☎ is the Fourier transform of , in modal coordinates. Can the Spatial Distribution of Damping be Measured? – p.5/22
✔ ✓ ✙ ☞ ✛ ☎ ✓ ✓ ✙ ☛ ✫ ✛ ☎ ✙ ✄ ☎ ✡ ✢ ✠ ✛ ✘ ✖ ✠ ✙ ✠ ✝ ✄ ✌ ✖ ✠ ✘ ☎ ✍ ☞ ✏ � ✍ � ✑ ✑ ✮ ✄ ✑ ✑ ✂ ✛ ✁ �✁ ✠ ✝ ☛ ✄ ☎ ✍ ✠ ✍ ☞ ✌ ☞ ☛ ✡ ✝ ✌ ✮ ✄ ✠ ✄ ✆ ✠ ✒ ✑ ☎ Non-viscous Damping Identification Damping model used for fitting: ✄✆☎ C Determine the complex natural frequencies, ✌✎✍ , and complex mode shapes, , from a set of measured ☎✝✆ transfer functions. Denote , ✔ ✟✞ ☎✎✍ . ✛ ✒✑ ☎✝✕ ☎✝✆ ☎✎✗ ☎✎✚ Set , and . ☎✎✜ M ☎✎✜ Obtain the relaxation parameter ☎✎✜ M ☎✣✢ Can the Spatial Distribution of Damping be Measured? – p.6/22
✑ ✄ ✖ ✠ ✁ ✬ � ✫ ☎ ✁ ✑ ✮ ✆ ✖ ✠ ✝ ✁ ✖ ✝ ✠ ✛ ✄ ✮ ✝ ✮ ✝ ✠ ☎ � ✮ ✑ ☎ ✄ ☎ ✠ ✬ ✂ ✄ � ✘ ☎ ✚ ✙ ✮ ✘ ✠ � ✙ ✮ ✑ ✮ ✁ ☎ � Non-viscous Damping Identification ☎ ✁� ☎✎✗ ☎✝✕ Obtain undamped modal matrix . ✄✆☎ ☎ ✁� ☎✎✚ Evaluate the matrix . ✄✆☎ ✄✆☎ ☎✝✕ ☎✝✕ ☎✝✕ From the matrix, get C and ☎✝✆ C ✔ ✟✞ ☎ ✁� ☎ ✁� ☎ ✁� ☎ ✁� Use to get the coefficient matrix in physical coordinates. Can the Spatial Distribution of Damping be Measured? – p.7/22
✠ ✟ �✁ ✝ ✠ ✂ ✝ ✟ ✄ ✠ ✠ ✟ ✝ ✂ ✁ ✡ ✞ ✗ ✂ ✝ ✝ ✄ ✠ ✞ Simulation Example m m k k k m k m k u u u u u u u u u . . . g(t) N− th g(t) Linear array of N spring-mass oscillators, , ✆✞✝ , . ✂☎✄ Simplest case: the kernel functions have the form ✄✆☎ ✄✆☎ C (3) ✄✆☎ is some damping function and C is a positive Here definite constant matrix. Can the Spatial Distribution of Damping be Measured? – p.8/22
� ✂ ✠ ✘ ☎ ✫ ✂ ✁ ✁ ✄ ✠ � ✄ ✆ ✝ ✁ ✂ ☎ ✝ ✝ ✁ ✍ � ✝ � ✑ ✁ ☛ ✝ ✠ ✑ ✫ ✁ ✁ ✂ ✦ � ✂ ✝ � � Models of Non-viscous Damping ✄✆☎ M ODEL 1 (exponential): ✄✆☎ M ODEL 2 (Gaussian): The damping models are normalized such that the damping functions have unit area when integrated to infinity, i.e ., ✄✆☎ Can the Spatial Distribution of Damping be Measured? – p.9/22
☞ ☎ ☞ ✁ ✝ ✁ ✂ ✁ ✠ � ✠ � ✡ ☛ ✆ ✂ ✝ ✄ ✝ ☎ ☎ ✂ ✠ � ✂ ✝ ✝ ✝ ✞ ✁ Characteristic Time Constant For each damping function the characteristic time ✄✆☎ constant is defined via the first moment of as ✄✆☎ Express as: . ✄✆☎ The constant is the non-dimensional characteristic time constant and is the minimum time period. ✄✆☎ Expect: near viscous ✂ ✟✞ strongly non-viscous Can the Spatial Distribution of Damping be Measured? – p.10/22
✘ ✁ ☛ ✁ ✠ ✁ Viscous Damping Identification 25 Fitted viscous damping matrix C kj 20 15 10 5 0 −5 30 25 20 30 15 25 20 10 15 5 10 5 0 0 j−th DOF k−th DOF Fitted viscous damping matrix for , damping model 2 Can the Spatial Distribution of Damping be Measured? – p.11/22
� ☛ ✁ ✠ ✁ Viscous Damping Identification 8 Fitted viscous damping matrix C kj 7 6 5 4 3 2 1 0 −1 −2 30 25 20 30 15 25 20 10 15 5 10 5 0 0 j−th DOF k−th DOF Fitted viscous damping matrix for , damping model 1 Can the Spatial Distribution of Damping be Measured? – p.12/22
✂✞ ✝ ✝ ✆ ✁ � � ☛ ✁ ✠ ✁ Non-viscous Damping Identification 30 25 Fitted coefficient matrix C kj 20 15 10 5 0 −5 −10 30 25 20 30 15 25 20 10 15 5 10 5 0 0 j−th DOF k−th DOF Fitted coefficient matrix of exponential model for , damping model 1; ✂☎✄ fit Can the Spatial Distribution of Damping be Measured? – p.13/22
� � � ✞ ✁ � � ☛ ✁ ✠ ✁ Non-viscous Damping Identification 12 10 Fitted coefficient matrix C kj 8 6 4 2 0 −2 −4 −6 30 25 20 30 15 25 20 10 15 5 10 5 0 0 j−th DOF k−th DOF Fitted coefficient matrix of exponential model for , damping model 2; ✂☎✄ fit Can the Spatial Distribution of Damping be Measured? – p.14/22
✩ ✂ ✍ ✄ � ✝ ✞ ✒ ✟ ✠ ✁ ✠ ✁ � ☛ Non-viscous Damping Identification −80 −90 Log amplitude of transfer function (dB) −100 −110 −120 −130 −140 original fitted using viscous −150 fitted using exponential −160 0 20 40 60 80 100 120 140 160 180 Frequency (Hz) Original and fitted transfer function ) for ✁✄✂ ( ☎✝✆ , damping model 2 Can the Spatial Distribution of Damping be Measured? – p.15/22
✂ ✄ � ✁ ✁ ✞ ✞ ✝ ✄ ✂ ✝ Experimental Setup Damped free-free beam: m, width = mm Clamped thickness = mm damping mechanism Instrumented hammer for impulse input Can the Spatial Distribution of Damping be Measured? – p.16/22
Measured Transfer Functions Driving Point Transfer Fucntion Measured 0 Reconstructed −10 −20 Log Amplitude dB −30 −40 −50 −60 −70 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Measured and fitted transfer function of the beam Can the Spatial Distribution of Damping be Measured? – p.17/22
Viscous Damping Fitting 6 x 10 5 Fitted viscous damping matrix, C kj 0 −5 −10 12 10 8 12 6 10 8 4 6 2 4 2 0 0 j−th DOF k−th DOF Fitted viscous damping matrix for damping between 4-5 nodes Can the Spatial Distribution of Damping be Measured? – p.18/22
✁ � ✂ ✞ ✄ ✁ ✁ � Non-viscous Damping Fitting 6 x 10 Coefficient of fitted exponential model, C kj 10 5 0 −5 12 10 8 12 6 10 8 4 6 2 4 2 0 0 j−th DOF k−th DOF Fitted coefficient matrix of exponential model for damping between 4-5 nodes; fit Can the Spatial Distribution of Damping be Measured? – p.19/22
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