Vibration Damping Professor Mike Brennan Vibration Control by Damping • Review of damping mechanisms and effects of damping – Viscous – Hysteretic – Coulomb • Modelling and characteristics of viscoelastic damping • Application of constrained and unconstrained layer damping • Measurement of damping Damping dissipation of mechanical energy (usually conversion into heat) • material damping (internal friction, mechanical hysteresis) • friction at joints • added layers of materials with high loss factors (e.g. visco- elastic materials) • hydraulic dampers (shock absorbers, hydromounts) • air/oil pumping: squeeze-film damping • impacts (conversion of energy into higher frequency vibration which is then dissipated) • acoustic radiation - vibration energy converted into sound (only important for lightly damped structures or heavy fluid loading, e.g. light plastic structures) • structural ‘radiation’ - transport of vibration into adjacent structures or fluids Review of Damping Mechanisms Viscous damping m x k c The equation of motion is mxxkcx 0 inertia damping stiffness force force force Example - automotive suspension system Free vibration effect of damping The underdamped displacement of the mass is given by nt x Xesind t Exponential decay term Oscillatory term = Damping ratio = cm2n 0 1 n = Undamped natural frequency = km 2 d = Damped natural frequency = n 1 = Phase angle Free vibration effect of damping nt x Xent x Xesind t time d 2 Damping ratio T d d Td Damping period Phase angle Hysteretic Damping Fe jt m x m x kj1 k c Loss factor Equation of motion mx k1 j x Fe jt mx cx kx Fe jt Assuming a harmonic response x Xe jt leads to X 1 X 1 F k2 m jk F k2 m j c Setting responses to be equal at resonance gives 2 where c2 mk Hysteretic damping force is in-phase with velocity and is proportional to displacement Viscous damping force is in-phase with and proportional to velocity Hysteretic Damping Hysteretic damping model with a constant stiffness and loss factor is only applicable for harmonic excitation f f kj1 m x m x k c causal acausal ft() ft() t xt() xt() Coulomb Damping Coulomb damping is caused by sliding or dry friction x xF +ve d k f xF -ve d m Fd = friction force (always opposes motion) Fd Fdn F x 0 0 x 0 F mg is the reaction force x n Fxn 0 • The damping force is independent of velocity once the motion is initiated, but the sign of the damping force is always opposite to that of the velocity. • No simple expression exists for this type of damping. We can equate the energy dissipated per cycle to that of a viscous damper How to compare damping mechanisms? Energy dissipated by a viscous damper Now if c x Xsin t Fd then x Xcos t x Substitute into (1) gives Fd cx 2 2 2 2 Energy lost per cycle is E c X cos t dt 0 E work done = force distance which evaluates to give T dx E F dx F dt dddt 0 E c X 2 2 E cx2 dt (1) 0 Returning to Coulomb Damping x k m f • Since the damping force Fd = Fn is constant and the distance travelled in one cycle is 4X, the energy dissipation per cycle is: EFX4 n 2 Noting that E ceq X 4F Gives an equivalent viscous damping coefficient c n eq X Note that this is a non-linear form of damping – dependent upon amplitude Returning to Coulomb Damping V log F V 4F log frequency • Illustration of the non-linear effects of Coulomb damping Coulomb Damping time • Illustration of the non-linear effects of Coulomb damping Equivalent viscous damping forces and coefficients Damping Damping Force Equivalent Mechanism Viscous Damping Viscous cx c k Hysteretic kx 4Fn Coulomb Fn X Forced vibration – effect of damping Fe jt X 1 F (k m 2 )2 (c)2 m x Frequency Regions k c Low frequency 0 X /F 1/ k Stiffness controlled Resonance 2 km X / F 1/c Damping controlled 2 2 High frequency n X /F 1/ m Mass controlled X Log F 1 k Stiffness Damping Mass controlled controlled controlled Log frequency Example: point mobility of 1.5 mm steel plate 0.6 0.5 m 0 10 = 0.01 -1 = 0.1 10 infinite plate -2 10 -3 10 Mobility Mobility (m/Ns) -4 10 -5 10 1 2 3 4 10 10 10 10 Frequency (Hz) • Damping is only effective at controlling the response around the resonance frequencies. • At high frequencies the mobility tends to that of an infinite structure and damping has no further effect. Visco-elastic materials • Visco-elastic materials (plastics, polymers, rubbers etc) have non- linear material laws. • For a harmonic input, visco-elastic materials are defined in terms of their complex Young’s modulus E(1+j) or shear modulus G(1+j). • Alternatively, we can write E = ER + jEI where ER is called the storage modulus and EI is called the loss modulus. • The parameters E, G and are dependent on – frequency – temperature – strain amplitude – preload • Frequency and temperature dependencies are equivalent (higher frequency is equivalent to lower temperature). A model with a constant modulus and constant loss factor is often a good approximation in a limited frequency region. However, it gives non-causal response in the time domain Typical material damping Visco-elastic damping – material properties k Simple model Voigt model • Combination of c c k Maxwell model k2 Dynamic stiffness is Fe jt 2 k2 k 1 k 2 2 j F c k k c 11 c k1 2 2 jt X 1 Xe ck1 Returning to Complex Stiffness kj( ) 1 ( ) F F kj( ) 1 ( ) X x F Stiffness determined from k( ) Re X F Im X Loss factor determined from () F Re X Visco-elastic damping – material properties Stiffness k 2 k k At low frequencies kk() 2 1 2 2 c2 k k k() 11 2 2 At high frequencies k() k k 1 12 ck1 kk12 logk ( ) k2 k1 Log frequency c Visco-elastic damping – material properties Loss factor c At low frequencies () k () c 2 k 2 k k 2 1 2 1 2 At high frequencies () c k11 k c k12 k kk11 log ( ) k1 Log frequency c Visco-elastic materials 4 10 Glassy Rubbery region 3 Transition region 10 region 2 10 Stiffness 1 10 0 10 Loss factor -1 Stiffness and loss factor loss and Stiffness 10 -2 10 Increasing frequency Increasing temperature Visco-elastic materials G or or or or E (a) (b) LogLog frequency frequency LogLog frequency frequency Visco-elastic materials – reduced frequency nomogram temperature η (Hz) f, and factor and loss E frequency modulus reduced frequency fαT, Hz Visco-elastic materials – reduced frequency nomogram • If the effects of damping and temperature on the damping behaviour of materials are to be taken into account, then the temperature- frequency equivalence (reduced frequency) is important. • E and η are plotted against the reduced frequency parameter, fαT. • To use the nomogram, for each specific f and Ti move down the Ti line until it crosses the horizontal f line. The intersection point X, corresponds to the value of fαT. Then move vertically up to read the values of E and η. Visco-elastic materials – reduced frequency nomogram Visco-elastic materials and damping treatments Useful range for damping treatments Modulus or loss factor Modulus Temperature • For damping treatments visco-elastic materials should generally be used in their transition region, where the loss factor is highest. However small changes in temperature can have a large change in stiffness. Typical material properties Material Young’s Density Poisson’s Loss factor modulus ρ (kg m-3) ratio ν η E (N m-2) Mild Steel 2e11 7.8e3 0.28 0.0001 – 0.0006 Aluminium Alloy 7.1e10 2.7e3 0.33 0.0001 – 0.0006 Brass 10e10 8.5e3 0.36 < 0.001 Copper 12.5e10 8.9e3 0.35 0.02 Glass 6e10 2.4e3 0.24 0.0006 – 0.002 Cork 1.2 – 2.4e3 0.13 – 0.17 Rubber 1e6 - 1e9 ≈ 1e3 0.4 – 0.5 ≈ 0.1 Plywood* 5.4e9 6e2 0.01 Perspex 5.6e9 1.2e3 0.4 0.03 Light Concrete* 3.8e9 1.3e3 0.015 Brick* 1.6e10 2e3 0.015 Damping treatments using visco- elastic materials Damping treatments using visco-elastic materials Viscoelastic damper Viscoelastic material Unconstrained layer damping Damping material Ejd (1 ) h2 h1 Structure Es Undeformed structure Deformed structure • Applied to one free surface • Loss due to extension of damping material (E product is important) • Frequency independent (apart from material properties) Unconstrained layer damping y Ejd (1 ) x Damping material Es h2 h1 Structure Undeformed structure Deformed structure • The strain energy in a uniform beam of length l is given by 2 1 l dy2 U EI dx (1) 2 2 0 dx Unconstrained layer damping • A definition of the loss factor is 1 energy dissipated per cycle 2 maximum energy stored per cycle Let b loss factor for the composite structure Let d loss factor for the damping layer Assume that the loss factor in the structure is negligible compared with the damping layer energy dissipated per cycle in the composite structure b d maximum energy stored per cycle in the composite structure energy dissipated per cycle in the damping layer maximum energy stored per cycle in the damping layer Unconstrained layer damping maximum energy stored per cycle in the damping l ayer So b d maximum energy stored per cycle in the composite structure Substitute from (1) gives EI b d d (I’s are about the composite neutral axis) dEIEI d d s s Thin damping layers EI b d d dEI s s Thus it is not sufficient to have a large ηd; a large Edηd is also required Thick damping layers bd To be effective damping layers of similar thickness to the structure are required Weight penalty Unconstrained layer damping 0 10 In this region the damping layer also affects the -1 Ed/Es=10 overall stiffness -1 10 of the structure.