Nonlinear Damping Mechanism andNonlinear Damping Mechanism and Vibration Isolation

Mike Brennan (UNESP)Mike Brennan (UNESP)Gianluca Gatti (University of Calabria, Italy)Bin Tang (Dalian University of Technology, China)g ( y gy, )

1

Introduction

Why to reduce vibration levels?y

High amplitude vibrations can cause fatigue and damage in machinery,

Excessive levels of noise from factories, vehicles and vibration transmitted through structures can

2

fatigue and damage in machinery, structures and sensitive equipment.

and vibration transmitted through structures can cause discomfort in humans.

Objetives

T th f f li i l ti• To compare the performance of a linear isolationsystem with the performance of that a nonlinearisolation system in which the damper is orientated atisolation system in which the damper is orientated atninety degrees with respect to the spring.

• To describe the way in which nonlinearity is generatedand then propagated through the nonlinear isolation

t i i th f t itt d th h thsystem examining the force transmitted through thespring and the damper.

3

Vibration Control Problem

A vibration control problem can be separated in:

Source ReceiverTransmission PathSource ReceiverTransmission Path

1. To reduce the vibrational excitation at source

2. To control the vibration by modifying the dynamic characteristics of the receiver to reduce its abilitycharacteristics of the receiver to reduce its ability of to respond.

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3. To isolate the receiver from the source placing an isolator (Vibration Isolation)

Vibration Isolation

5

Vibration Isolation

There are two common problems:

Isolate a vibrating machine from its surrounding

Isolation from a vibrating host structure or basefrom its surrounding host structure or base

6

A SDOF Linear Isolator

j tX e

j teF e

equipmentm

j ttX e

Sourcem j tX

Isolatork c

Isolatork c

j tXe

k cj t

eY ek c

BaseReceiverj ttF e

Base Excited SystemForce Excited System

• Linear Spring

7

p g• Linear Viscous Damper• Harmonically force/base Excited

Performance of an Isolator

The Equations of Motion

e ︵ ︶mx cx kx f t

mz cz kz my z x yForce Excited

Base Excited

When the system is excited by a harmonic force:

Source

j teF e

j tXe mTransmitted Force

t ︵ ︶F k j c X

Excitation Force 2 e ︵ ︶F k m j c XIsolator k c

Force Transmissibility

e ︵ ︶j

tF k j cT

j ttF e

Receiver

Force Transmissibility 2Fe

TF k m j c

Di T i ibilit tX k j cT

8

Disp. Transmissibility 2

t

eD

jTY k m j c

Transmissibility 2 n

,cm

n

2

1 21 2

F D jT T

j1 2 j

Transmissibility (in dBs)

At Frequencies 1

1020

t

e

︵ ︶ log FT dBF

At Frequencies   1

1 0 , ︵ ︶T T dB

11T

At the resonance Frequency  

1 2T

In the Isolation RegionHigh damping T reduces at 20 dB/decade

91

2 T

1 2d / High damping T reduces at 20 dB/decadeLow damping T reduces at 40 dB/decade

Nonlinear Damper

fe f

20

Isolation RegionIsolation Region

Om m

O

fe

ty| (

dB)

20

0gg

c3k

c 1x

kx

O

c1

ch

nsm

issi

bilit -20

-403c1

|Tra

n-60

d

a yft yftNon-dimensional frequency

.1 1 10 100-80

•• Linear dampingLinear damping•• √ Resonance√ Resonance•• ×× High frequencyHigh frequency

Cubic dampingCubic damping√ √ Force transmissibilityForce transmissibility×× Displacement transmissibilityDisplacement transmissibility√ √ High frequencyHigh frequency

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√ Force and disp. transmissibility√ Force and disp. transmissibility

Nonlinear Damping Isolator

fe 2

221

h e

ˆ ˆˆ ˆ ˆ'' ' cosˆ

xx x x Fx

Om

fe

22

221

h

ˆ ˆˆ ˆ ˆ'' ' cosˆ

zz z z Yzk

c 1

O

xch 1 z

n

ˆ ' ,xxa

2n

ˆ '' xxa

kh

Numerical Methods are necessaries to solve them

n

a yftnecessaries to solve them

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Low Amplitude ExcitationFor Amplitudes of excitation so that: 0 2.z a

2z eq h

2x eq h eq h

The Non‐dimensional Equations of Motion

ˆˆ ˆ ˆ 2 ˆˆ ˆ ˆ2 eq eˆ ˆ ˆ'' ' cosx x x FForce Excited System Base Excited System

22 eqˆ ˆ ˆ'' ' cosz z z Y

122n

ˆ '' xxa

n

ˆ ' ,xxa

State Space The damped unforced oscillator is

22 0 ˆ ˆ ˆ ˆ'' 'x x x x

which can also be called as an autonomous system,

22 0 h'' 'x x x x

y ,or time-invariant systems.

2

︵ ︶d x t 2

︵ ︶d x t 2

2

︵ ︶ ︵ ︶ ︵ ︶

d x t f x t x tdt

, 2

2

︵ ︶ ︵ ︶ ︵ ︶

d x t g x t x t tdt

, ,

Introducing an variableˆ ˆ 'y x

22ˆ ˆ ˆ ˆ'y x x x h

y x

22 0 hˆ ˆ ˆ ˆ'' 'x x x x

13

hh22ˆ ˆ ˆ ˆy x y x hor

Phase Plane

0 1ˆ ˆ

x xd22 0 ˆ ˆ ˆ ˆ'' 'x x x x 2ˆ ˆˆ1 2

hy yd x2 0 hx x x x

q qf

The damped unforced oscillator can be present in a

q qf

The damped unforced oscillator can be present in a phase plane, for example

ˆ ˆq x y

As time passes the points describe a curve in

q x y

ˆ ˆx yp pthe phase plane, which is called an orbit, a trajectory, or and integral curve.

y

14

Phase Plane 2

0 1ˆ ˆˆ ˆˆ1 2

h

x xdy ydt x

ˆ ˆq x y

2ˆ ˆ ˆˆ ˆ 2 hx x yy dy d

Integrating the equation analytically/numerically

ˆ ˆ ˆx dx d y

g g q y y y0.2 0.2

0.1 0.1

0.0 0.0x′ ˆ x′ ˆ

-0.1 -0.1

15 -0.2 -0.1 0.0 0.1 0.2

-0.2

-0.2 -0.1 0.0 0.1 0.2

-0.2

x x

Singular Point

q qf

When q' = 0, the points on the phase plane are termed singular points, fixed pint, equilibriums or zeros. Physical meaning. Singular Point and stability:g y

Node; Saddle;; Focus; Center

16

Singular Point (From S.S. Rao Mech. Vib.)

q qf

17

Free Vibration Characteristics

18

Low Amplitude ExcitationFor Amplitudes of excitation so that: 0 2.z a

214eq h Z 4

The Non‐dimensional Equations of Motion

ˆˆ ˆ ˆ'' ' 2 ˆˆ ˆ ˆ'' '2 eq e'' ' cosx x x FForce Excited System Base Excited System

22 eq'' ' cosz z z Y

192n

ˆ '' xxa

n

ˆ' ,xxa

Low Amplitude Transmissibility

Force and Displacement Transmissibility are not the same

2112 h

F

ˆj XT

2

h112

ˆj ZT

F T i ibili

22 112

F

hˆ

Tj X

Di l T i ibili

D 22h

112

ˆT

j Z

Force Transmissibility Displacement Transmissibility

F 2 ˆY22 11

2 h

ˆˆ

eFXj X

2

22 112 h

ˆˆ

YZj Z

20

Low Amplitude Transmissibility

For the maximum allowable amplitude for Force and Baseamplitude for Force and Base Excitation:  

0 4F 0 4ˆY 0 4e .F 0 4.Y

And

10h

And

0 1L .

21

Results

22

High Amplitude Transmissibility

The Equations are solved Numerically

2

221

h e

ˆ ˆˆ ˆ ˆ'' ' cosˆ

xx x x F21h ex

2ˆ ˆz 222

1

hˆˆ ˆ ˆ'' ' cos

ˆzz z z Yz

After the transient, the maximum values from the steady state response at the frequency are used to plot the forcestate response at the frequency  are used to plot the force and displacement transmissibilities

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Force Transmissibility (High Amplitude Excitation)

Using:0 1L . , 10h

Using:

For: 0 1e 0.2, 0.3, 0.5. ˆ . ,F e , ,,

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Displacement Transmissibility (High Amp. Excit.)

Using:0 1L . , 10h

Using:

For: 0 1e 0.2, 0.3, 0.5. ˆ . ,Y e

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Detailed Analysis of the Nonlinear System

The frequency region of interest is roughly 

0 3 1 6. .

The system is analysed at:

0 5 1 0. , .

For:  0 5ˆ .eF0.3 1.6

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Procedure

2ˆFor the Force Excited System

2

221

h eˆ ˆ ˆ'' ' cosˆ

xx x x Fx

F Th h h S iForce Through the Spring Force Through the Damper

2

2ˆ ˆ 'xF ˆF 22

1d h 'ˆ

F xx

k ˆF x

li i C ffi iTransmitted Force

2x

Nonlinear Damping Coefficient

2x22

1

t h ˆ ˆ'ˆ

xF x xx 22

1f h ˆxcx

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The Fourier Series

A periodic signal can be represented by adding together sineand cosine functions of appropriated frequencies,pp p q ,amplitudes and relative phases.

0 2 2︵ ︶ cos sina nt ntx t a b

0

12 n nn︵ ︶ cos sinx t a b

T T

2 2Amplitude and Phase

2 2

1

n

n n ntan ︵ / ︶

n nA a bb a

Fourier Coefficients0

0

12 ︵ ︶

Ta x t dtT

0

2 2n n = 1, 2, 3,... ︵ ︶cos

T

T

nta x t dtT T

280

2 2n ︵ ︶sin

T ntb x t dtT T

Transmitted Force2

221

t h

ˆ ˆ ˆ'ˆ

xF x xx

0 5.

1 0 .

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Force Through the Spring k ˆF x

0 5.

1 0 .

30

Force Through the Damper2

221d h

ˆ ˆ 'ˆ

xF xx

0 5.

1 0 .

31

Nonlinear Damping Coefficient2

221f h

ˆˆ

xcx

0 5.

1 0 .

32

Velocity of the Mass ˆ 'x0 5.

1 0 .

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Concluding Remarks

The nonlinear damper is suitable suitable for usein a vibration isolator with low amplitudein a vibration isolator with low amplitudeexcitation.

When the excitation is large then the systembecomes highly nonlinear at frequencies close tobecomes highly nonlinear at frequencies close tothe resonance frequency of the system.

The damping nonlinearity results in distortion ofthe velocity of the suspended mass and thisthe velocity of the suspended mass and thiscombines with time varying nature of the dampingin the system

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in the system.

References[1] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations. Wiley, New York, 1995.[2] J.J. Thomsen, Vibrations and Stability, Advanced Theory, Analysis, and Tools, 2nd ed., Springer, Berlin, 2003.[3] N. Kryloff, N. Bogoliuboff, Introduction to Non-linear Mechanics, Princeton University Press, Princeton, 1943. [4] J.E. Ruzicka, T.F. Derby, Influence of Damping in Vibration Isolation, The Shock and Vibration Information Center Washington D C 1971Center, Washington, D.C., 1971.

[5] R E Mickens Analytical and numerical study of a non[5] R.E. Mickens, Analytical and numerical study of a non-standard finite difference scheme for the unplugged van derPol equation, Journal of Sound and Vibration, 245, 2001,

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Pol equation, Journal of Sound and Vibration, 245, 2001, 757-761.

References[6] G.N. Jazar, R. Houim, A. Narimani, M.F. Golnaraghi, Frequency response and jump avoidance in a nonlinear passive engine mount, Journal of Vibration and Control12(11) (2006) 1205-1237.

[ ] Z Q L X J Ji S A Billi G R T li Z K[7] Z.Q. Lang, X.J. Jing, S.A. Billings, G.R. Tomlinson, Z.K. Peng, Theoretical study of the effects of nonlinear viscous damping on vibration isolation of sdof systems Journal ofdamping on vibration isolation of sdof systems, Journal of Sound and Vibration 323(1-2) (2009) 352-365.[8] Bin Tang M J Brennan A comparison of the effects of[8] Bin Tang, M.J. Brennan. A comparison of the effects of nonlinear damping on the free vibration of a single-degree-of-freedom system, Transactions of the ASME, Journal of Vibration and Acoustics 134(2) (2012) 024501.[9] Bin Tang, M.J. Brennan. A comparison of two nonlinear

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damping mechanisms in a vibration isolator, Journal of Sound and Vibration 332(3) (2013) 510-520.

References[10] J.C. Carranza, Comparison between the Performances of a Linear Isolator and an Isolator with a Geometrically Nonlinear Damper, M.Sc. Dissertation, UNESP Ilha Solteira, 2013.

[11] J C C M J B B T A l i f[11] J.C. Carranza, M.J. Brennan, B. Tang. Analysis of a geometrically nonlinear damping mechanism in a vibration isolator The Fourteenth Pan American Congress of Appliedisolator, The Fourteenth Pan American Congress of Applied Mechanics (PACAM XIV), Santiago, Chile, March 24-28, 2014.

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Thank You for Your Attention!!!Thank You for Your Attention!!!Any Questions are welcome!y

谢谢 (Xièxiè)！( )

Bin Tang

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Institute of Internal Combustion Engine, Dalian University of Technology, China.