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Vibration

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 (usually conversion into heat)

• material damping (internal , 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 vibration which is then dissipated) • acoustic radiation - vibration energy converted into (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 is mxxkcx   0

damping stiffness force force Example - automotive suspension system Free vibration effect of damping

The underdamped of the is given by

nt x Xesind t 

Exponential decay term Oscillatory term

 = Damping ratio = cm2n   0 1

n = Undamped natural frequency = km

2 d = Damped natural frequency = n 1  = angle Free vibration effect of damping

x Xent

nt d x Xesind t 

2   Damping ratio T  d  d Td  Damping period   Phase angle Hysteretic Damping Fe jt

m x m x kj1  k c Loss factor Equation of motion mx k1  j x  Fe jt mx cx  kx  Fe jt Assuming a harmonic response x  Xe jt  leads to X 1 X 1   F k2 m jk F k2 m j c Setting responses to be equal at gives  2 where   c2 mk

Hysteretic damping force is in-phase with 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

m x m x k c kj1  causal acausal

ft() ft()

t xt() xt() Coulomb Damping Coulomb damping is caused by 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  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 f m

• 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 4F Gives an equivalent viscous damping coefficient c  n eq  X

Note that this is a non-linear form of damping – dependent upon 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 and coefficients

Damping Damping Force Equivalent Mechanism Viscous Damping

Viscous cx c

k Hysteretic kx 

4Fn Coulomb Fn  X Forced vibration – effect of damping jt Fe 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 . • 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 – – 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   ck1 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   ck1

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 region 3 Rubbery

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 E

(a) (b)

LogLog frequency frequency LogLog frequency frequency Visco-elastic materials – reduced frequency nomogram

temperature

η

(Hz)

f,

and loss factor loss and

E 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 loss or 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

x Damping material Ejd (1  )

h2

h1 Structure Es

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. c b

d E /E =10-2  1 d s b  i.e. require: -2  EIss 10 d 1 Ed large -3 E /E =10 EIdd d s h2 large

d large -4 Ed/Es=10 -3 10 -1 0 1 2 h2 10 10 10 10 h1 Constrained layer damping

Constraining layer Damping material Gj(1  ) E3 h3 h d 2 h 1 Structure E1 Undeformed Deformed constrained layer constrained layer

• Sandwich construction

• Loss due to shear of damping layer (G product is important)

• Constraining layers must be stiff in extension

• Thin damping layers can give high losses

• Frequency dependent properties Constrained layer damping

composite/ depends on two non-dimensional parameters:

• ‘geometric parameter’ (also depends on elastic moduli)

2 h : h : h L E h() E h d 1 1 3 L  1 1 3 3 1 : 0.1 : 0.1 0.46 E1 h 1 E 3 h 3 E 1 I 1 E 3 I 3  1 : 0.1 : 1 3.63 1 : 1 : 1 12 1 : 10 : 1 363 • here I1 and I3 are relative to own centroids • large for deep core, small for thin core (d) (E1=E3) • ‘shear parameter’ G 11 g 2  hk2 b E1 h 1 E 3 h 3

Bending wavenumber

• large for low frequencies, small for high frequencies Constrained layer damping

0 10 In this region L=100 large L means much stiffer -1 10 L=10 structure  composite L=1

-2  10

L=0.1

-3 2 10 E h() E h d L  1 1 3 3 E1 h 1 E 3 h 3 E 1 I 1 E 3 I 3 

-4 10 -2 -1 0 1 2 10 10 10 10 10 G 11 Shear parameter g 2  hk2 b E1 h 1 E 3 h 3 Constrained layer damping

Effectiveness also depends on loss factor of damping material

At low frequencies (high temp) constrained layer damping is ineffective because the damping material is operating in its rubbery region and is thus soft. The structure and the constraining layer become uncoupled.

At high frequencies (low temp) constrained layer damping is ineffective because the damping material is operating in its glassy region and is thus hard. The structure and the constraining layer tend to move as one. Other types of damping

Damping by air pumping

Viscous damping – reduces in effectiveness at high frequencies

Fe jt Pumping along joints Squeeze-film damping I

Attached plate Thin layer of air Main plate

Damping is achieved by air being pumped between the main plate and the attached plate (JSV 118(1), 123-139, 1987) Squeeze-film damping II

Oil supply

Stationary casing Oil film

vibration Rotating shaft

Ball/roller bearing

• Oil film is squeezed between the bearing and the housing generating damping forces Summary: means of adding damping

• unconstrained layer damper

• constrained layer damper (can be multiple layers)

• friction damping: important at bolted or rivetted joints.

• squeeze film damping

• impact damper

• change in structural material.

• tuned vibration absorber (added damped mass-spring systems) Measurement of Damping

• Decay of free vibration measurement

• Response curve at resonance measurement

• Complex modulus measurement Measurement of Damping –decay of free vibration xt  Measure the amplitude nt x Xe of two successive cycles x 1  endT x2 x1 x2 t x1 ln ndT x2 Logarithmic decrement 2 So   1  2 t1 t21 t Td For light damping  2  22 T   measured d 2 d 1 So   n 2 Measurement of Damping – decay of free vibration

• In practice it is better to measure the logarithmic decrement over a number of cycles

1 xi   ln nxin

n can be determined by knowledge of the natural frequency

and the time between xi and xi+n Measurement of Damping – response curve at resonance X F Response curve Nyquist plot 1 k Xmax F  2 X Re  F

Xmax 2F

 X n Im F  1 2 frequency n    21 2n Measurement of Damping – complex modulus measurement

shaker Impedance head (measures force and ) very stiff, light material specimen l area A

F F Measure dynamic stiffness kj(1 ) kj(1  ) X X F Im F X Stiffness k  Re  Loss factor   X F Re  EA X Compute E by using k  l Summary

• Review of damping mechanisms

• Modelling and characteristics of viscoelastic damping

• Application of damping treatments

• Measurement of damping

References

• A.D. Nashif, D.I.G. Jones and J.P. Henderson, 1985, Vibration Damping, John Wiley and Sons Inc. • C.M. Harris, 1987, Shock and Vibration Handbook, Third Edition, McGraw Hill. • L.L. Beranek and I.L. Ver, 1992. and , John Wiley and Sons. • D.J. Mead, 1999, Passive Vibration Control, John Wiley and Sons. • F.J. Fahy and J.G. Walker, 2004, Advanced Applications in , Noise and Vibration, Spon Press (chapter 12 Vibration Control by M.J. Brennan and N.S. Ferguson). • F.S. Tse, I.E. Morse and R.T. Hinkle, 1978, Mechanical Vibrations, Theory and Applications, Second Edition, Allyn and Bacon, Inc.