Vibration Isolation. Gianluca Gatti (University of Calabria, Italy) Bin Tang (Dalian University of Technology, China)

Transcription

1 Nonlinear Damping Mechanism and Vibration Isolation Mike Brennan (UNESP) Gianluca Gatti (University of Calabria, Italy) Bin Tang (Dalian University of Technology, China) 1

2 Introduction Why to reduce vibration levels? Excessive levels of noise from factories, vehicles and vibration transmitted through structures can cause discomfort in humans. High amplitude vibrations can cause fatigue and damage in machinery, structures and sensitive equipment.

3 Objetives To compare the performance of a linear isolation system with the performance of that a nonlinear isolation system in which the damper is orientated at ninety degrees with respect to the spring. To describe the way in which nonlinearity is generated and then propagated through the nonlinear isolation system examining i the force transmitted throughh the spring and the damper. 3

4 Vibration Control Problem A vibration control problem can be separated in: Source Transmission Path Receiver 1. To reduce the vibrational excitation at source. To control the vibration by modifying the dynamic characteristics of the receiver to reduce its ability of to respond. 3. To isolate the receiver from the source placing an isolator (Vibration Isolation) 4

5 Vibration Isolation 5

6 Vibration Isolation There are two common problems: Isolate a vibrating machine from its surrounding Isolation from a vibrating host structure or base 6

7 A SDOF Linear Isolator Fe e k jt Source m Xe Isolator c j t equipment m k Isolator c j t Xe t Ye e jt Fe t jt Receiver Base Force Excited System Base Excited System Linear Spring Linear Viscous Damper Harmonically force/base Excited 7

8 )))Performance of an Isolator The Equations of Motion mxcxkxf t e (zmczkzmbase Excited y zwhen the system is excited by a harmonic force: Transmitted Force Excitation Force t (Excited xyforce FFkjcX kmj e (Force Transmissibility kf t Disp. Transmissibility ibilit k c Xk j t Fe t Source m j t Fe e c j t Xe Isolator Receiver kjcttf mjef c kjcttx YmjceD 8

9 )()TTransmissibility m n,jt1 F D 1 j1 c n Transmissibility (in dbs) FTdBlog t 0 10 ( e At Frequencies 1 FAt TTd1 B 0,At the resonance Frequency 1T 1 In the Isolation Region T /1 1 d High damping T reduces at 0 db/decade Low damping T reduces at 40 db/decade 9

10 Nonlinear Damper 0 c 1 c h c 1 m f e k Tran nsmissibilit ty (db) O 0-0 x c 3 m Isolation f Region f e k d O x a f t Linear damping Resonance High frequency Force and disp. transmissibility -80 y y f.1 1 t Non-dimensional frequency Cubic damping Force transmissibility Displacement transmissibility 10

11 Nonlinear Damping Isolator c 1 c h a m f e k f t O xo x y ˆx'xˆx'ˆxˆFcos ˆ h eˆ1 ˆ h 1 zk ˆzˆYcˆz'ˆz'ˆzos'xnˆ x, xx' a a nˆnumerical Methods are necessaries to solve them n 11

12 a 'Low Amplitude Excitation zfor Amplitudes of excitation so that: 0. eq eq ˆz ˆx h h The Non dimensional Equations of ˆxˆxˆxMotionˆˆeq eˆfforce Excited System Base Excited System ''ˆcosˆz'ˆˆeqˆz'ˆz ˆ ˆcos Y 'x x,a nˆx xa nˆ1

13 )()())()()State Space The damped unforced oscillator is x'ˆxˆx'ˆ ˆ 0 h which can also be called as an x autonomous system,, or time-invariant systems. dx dt( t gxtxtt,,, tdxfxtxt dt( (Introducing an variableˆx'ˆˆˆ hˆ yˆx' y ˆy ˆˆxˆx'ˆ yˆ xx'x 0 x h h ˆxˆ or x h 13

14 Phase Plane x'ˆxˆx'ˆxd xˆ 0 1 ˆ0 h d yˆ 1 hxˆ q f q xˆ yˆ The damped unforced oscillator can be present in a phase plane, for example ˆ ˆ q x y As time passes the points xˆ y ˆ describe a curve in the phase plane, which is called an orbit, a trajectory, or and integral curve. 14

15 xˆ yˆ 0 1 Phase Plane d dt 1 hxˆ xˆ y ˆ q xˆ yˆ yˆ dyˆ d xˆ hxˆ yˆ xˆ dxˆ d yˆ Integrating g g the equation analytically/numerically y y x ˆ x ˆ ˆx ˆx 15

16 Singular Point q f q When q' = 0, the points on the phase plane are termed singular points, fixed pint, equilibriums or zeros. Physical meaning. Singular Point and stability: Node; Saddle; Focus; Center 16

17 Singular Point (From S.S. Rao Mech. Vib.) q f q 17

18 Free Vibration Characteristics 18

19 a Low Amplitude Excitation zfor Amplitudes of excitation so that: 0.1 eq Z 4 hˆ The Non dimensional Equations of ˆx'ˆx'ˆxosMotionˆˆz'ˆzˆzcos ' ˆ eq ec eq Excited System FForce Base Excited System Y 'x x,anˆ'nˆx xa 19

20 Low Amplitude Transmissibility Force and Displacement Transmissibility are not the same TF 1 jˆx1 h 1 jx1 hˆ F T i ibili Force Transmissibility Displacement Transmissibilityˆ T1 jˆz1 h 1 jˆz1 h D Di l T i ibili X hˆ1 FˆYeZ hˆˆ 1 jx 1 j 1 ˆZ 0

21 Low Amplitude Transmissibility For the maximum allowable amplitude for Force and Base ˆF.ˆ 4 Y0 e 04. Excitation: And h 10 L. 01 1

22 Results

23 ˆ'High Amplitude Transmissibility The Equations are solved Numerically ˆx'x ˆx'ˆxˆcos ˆ h F eˆ1 h exˆzz ˆz'ˆzˆYcos ˆ h 1 zafter the transient, the maximum values from the steady state response at the frequency are used to plot the force and displacement transmissibilities 3

24 Force Transmissibility (High Amplitude Excitation) Using:.L 01, For: F 10 h ˆ., 01 0., 0.3, 0.5. e 4

25 Displacement Transmissibility (High Amp. Excit.) Using:.L 01, 10 For: ˆY., 01 0., 0.3, 0.5. e h 5

26 Detailed Analysis of the Nonlinear System The frequency region of interest is roughly The system is analysed at:., eFor: 05.F ˆ6

27 Procedure the Force Excited System ˆFor xˆx' ˆx'ˆxFcos ˆ h eˆ1 Force Through h the Spring xf Force Through the Damper ˆxFxF ˆ'x ˆˆx k d h 1 Transmitted Force ˆxFˆ ˆt h 1 Nonlinear Damping Coefficient i x'ˆxcˆx ˆx f h x 1 7

28 )(td))))the Fourier Series A periodic signal can be represented by adding together sine and cosine functions of appropriated p frequencies, amplitudes and relative phases. antntxt acos sinb 0 T Amplitude and Phase AFourier Coefficients 0 n n ( 1 T n (n n n nt an ab n nb/1 a at0 1 xtt ( 0 Tnta xtcos TTn = 1,, 3,... n (0 Tntb xtsin TTn ( 0 dtdt8

29 5. x0 Transmitted Force Fˆx ˆt h 1 ˆx'ˆ x 10.9

30 05. FˆForce Through the Spring k x 10.30

31 05. Force Through the Damper Fˆx ˆd h 1 x ˆ'x 10.31

32 05. Nonlinear Damping Coefficient cˆx ˆf h 1 x 10.3

33 ˆ'x0 5. Velocity of the Mass.10 33

34 Concluding Remarks The nonlinear damper is suitable suitable for use in a vibration isolator with low amplitude excitation. When the excitation is large then the system becomes highly nonlinear at frequencies close to the resonance frequency of the system. The damping nonlinearity results in distortion of the velocity of the suspended mass and this combines with time varying nature of the damping in the system. 34

35 References [1] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations. Wiley, New York, [] J.J. Thomsen, Vibrations and Stability, Advanced Theory, Analysis, and Tools, nd ed., Springer, Berlin, 003. [3] N. Kryloff, N. Bogoliuboff, Introduction to Non-linear Mechanics, Princeton University Press, Princeton, [4] J.E. Ruzicka, T.F. Derby, Influence of Damping in Vibration Isolation, The Shock and Vibration Information Center, Washington, D.C., [5] R.E. Mickens, Analytical and numerical study of a nonstandard finite difference scheme for the unplugged van der Pol equation, Journal of Sound and Vibration, 45, 001,

36 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 Control 1(11) (006) [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 of Sound and Vibration 33(1-) (009) [8] Bin Tang, M.J. Brennan. A comparison of the effects of nonlinear damping on the free vibration of a single-degreeof-freedom system, Transactions of the ASME, Journal of Vibration and Acoustics 134() (01) [9] Bin Tang, M.J. Brennan. A comparison of two nonlinear damping mechanisms in a vibration isolator, Journal of Sound and Vibration 33(3) (013)

37 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, 013. [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 Applied Mechanics (PACAM XIV), Santiago, Chile, March 4-8,

38 Thank You for Your Attention!!! Any Questions are welcome! 谢谢 (Xièxiè)!( ) Bin Tang Institute of Internal Combustion Engine, Dalian University of Technology, China. 38