Methods of Modeling Damping in Flexible Structures Michael Siegert, Alexander Lion, Yu Du Institute of Mechanics Faculty of Aerospace Engineering Universität der Bundeswehr München Volkswagen AG Wolfsburg 23rd May 2014
Content 1 Motivation 2 Experiments 3 Modeling approach 4 Conclusions
Content 1 Motivation 2 Experiments 3 Modeling approach 4 Conclusions
What is damping? Damping... is the dissipation of energy by vibrating systems. Damping has to be separated from: Isolation... avoids vibrations to enter or to leave a system, ˆx Absorption... is achieved by counter-vibrating masses mounted on the excited structure, such that the excited structure itself is not vibrating. ω
Why is modeling of damping important? Continuous increase of simulation in the product development phase dynamic vehicle behavior MBS flexible structures FEM damping? comfort analysis Damping may have major impact on the simulation results!!! Remark: Up to date, it is not possible to determine structural damping without experimental investigations.
Mechanisms of damping Material damping Viscoelasticity Heat flux Dislocation climb... Joint damping Microslip Gas-pumping / Squeeze-film-damping... Fluid-structure-interaction with surrounding medium Viscous damping Sound radiation... Coatings etc...
Content 1 Motivation 2 Experiments 3 Modeling approach 4 Conclusions
Experimental setup Specimen Details: Material: V2A Length l [mm]: 400 Width b [mm]: 15, 50, 100, 150, 200 Thickness t [mm]: 1.5 t 00 11 00 11 00 11 00 11 Setup Details: Distance d l in [mm] between Loudspeaker and Specimen : 10, 20, 40, 80, 160, 320 Polytec OFV 302 b d l Loudspeaker 00000000000000000000 11111111111111111111 Polytec OFV 3000 l Specimen Amplifier v Data Acquisition to PC Sine Wave Generator
Signal analysis reliable signal amplitude via 2a rms rms-value: 1 T2 a rms = [a(t)] T 2 T 2 dt. (1) 1 T 1 damping constant δ via logarithmic decrement Λ ( ) δ = Λ ln a(t1 ) T = a(t 2 ) (2) T 2 T 1 50 45 40 35 30 25 20 15 10 5 0 5 10 15 20 25 30 35 40 45 50
Amplitude decay Velocity in [mm/s] 10 3 10 2 10 1 linear b = 15mm, d = 20mm l nonlinear b = 15mm, d l = 20mm linear b = 50mm, d l = 20mm nonlinear b = 50mm, d = 20mm l linear b = 100mm, d l = 20mm nonlinear b = 100mm, d l = 20mm linear b = 150mm, d = 20mm l nonlinear b = 150mm, d l = 20mm linear b = 200mm, d = 20mm l nonlinear b = 200mm, d l = 20mm 10 0 10 1 0 10 20 30 40 50 60 70 80 time in [s]
Damping over amplitude Decay Constant (δ) in [1/s] 10 1 b = 15mm, d = 10mm l b = 15mm, d l = 40mm b = 50mm, d l = 10mm b = 50mm, d l = 40mm b = 50mm, d l = 160mm b = 100mm, d = 10mm l b = 100mm, d l = 40mm b = 100mm, d = 160mm l b = 150mm, d l = 10mm b = 150mm, d = 40mm l b = 150mm, d l = 160mm b = 200mm, d l = 40mm b = 200mm, d l = 160mm 10 0 10 1 10 2 10 3 Velocity Amplitude (v) in [mm/s]
Content 1 Motivation 2 Experiments 3 Modeling approach 4 Conclusions
Euler-Bernoulli beam theory Euler-Bernoulli beam theory: ρ S A 2 w(x,t) t 2 +EI yy 4 w(x,t) x 4 = 0. (3) Displacement w(x, t) in free-free boundary condition: w(x,t) = [C 1,j (cos(λ j x)+cosh(λ j x))+c 2,j (sin(λ j x)+sinh(λ j x))]e i(ωt+ϕ 0), (4) C 1,j = C 2,j sinh(λ j ) sin(λ j ) cosh(λ j ) cos(λ j ). (5)
Results for 1st bending mode 1st bending modes of specimens dependent on width: specimen width b w(x) in [mm] 1 0.5 0 0.5 48.6419 Hz 1 0 50 100 150 200 250 300 350 400 x in [mm] 15 mm 48.35 Hz 48.91 Hz 50 mm 48.20 Hz 48.88 Hz 100 mm 47.61 Hz 48.79 Hz 150 mm 47.36 Hz 48.69 Hz 48.64 Hz 200 mm 47.45 Hz 48.64 Hz
Simplified Navier-Stokes equations Balance of mass: ˆρ F +ρ F,0 u = 0. (6) t Navier-Stokes-equations: [ ] û ρ F,0 t +(u 0 )(û) = ˆp +η û+k, (7) ρ F (x,t) = ρ F,0 + ˆρ F (x,t), (8) p(x,t) = p 0 + ˆp(x,t), (9) u(x,t) = u }{{} 0 +û(x,t). (10) =0
Acoustic potential and dipole 10 The acoustic dipole is given by the acoustic potential Φ(r, t), 0 10 Φ(r,t) = Φ 0 cos(θ) 1+ikr r 2 e i(ωt kr). (11) 0.05 0.00 It offers the velocity of air movement, 0.05 10 0 10 0.08 û(r,t) = Φ(r,t) (12) 0.06 0.04 and the air pressure, 0.02 0.02 5 10 15 20 r in [m] ˆp(r,t) = ρ F,0 Φ t. (13) 0.04 v r (r,θ = 0) p(r,θ = 0)
Extension of Euler-Bernoulli beam theory Euler-Bernoulli beam will be extended by a viscous term: ρ S A d2 w(x,t) dt 2 +EI yy d 4 w(x,t) dx 4 = q(x,t)ẇ(x,t). (14) Additional linearly distributed load is from the acoustic dipole model: q(x, t) =? (15)
Content 1 Motivation 2 Experiments 3 Modeling approach 4 Conclusions
Conclusions & upcoming challenges Negligible influence of specimen width in linear damping Effect of air inertia is higher than its damping properties, but is still negligible in comparison to the inertia properties of steel Damping due to Fluid-Structure-Interaction depends highly on the surrounding environment Stochastic damping model for unknown environments or fixed test case?
Further work Extension of the model to a broader frequency width Including fluid boundary conditions in fluid-structure-damping-model Testing of the fluid-structure-model in hypobaric chambers Considering complex geometry designs in fluid-structure-damping Developing a corresponding FE model Modeling further damping mechanisms, other than fluid-structure-damping
References Fahy, F. (1985). Sound and Structural Vibration. Academic Press. Guyader, J.-L. (2006). Vibration in Continuous Media. ISTE. Vanwalleghem, J., I. D. Baere, M. Loccufier, and W. V. Paepegem (2014). External damping losses in measuring the vibration damping properties in lightly damped specimens using transient time-domain methods. Journal of Sound and Vibration 333(6), 1596 1611. Walker, S. J., G. S. Aglietti, and P. Cunningham (2009). A study of joint damping in metal plates. Acta Astronautica 65(1 2), 184 191.