Mobility and Impedance Methods. Professor Mike Brennan

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1 Mobility and Impedance Methods Professor Mike Brennan

2 ibration control ibration Problem Understand problem Modelling (Mobility and Impedance Methods) Solve Problem Measurement

3 Mobility and Impedance The response of a structure to a harmonic force can be expressed in terms of its mobility or impedance General linear system At frequency the velocity can be written in complex notation v(t) = e jt is the complex amplitude. Similarly for the force f(t) = e jt The mobility is defined as j Mobility j The impedance is defined as Impedance j j If the force and velocity are at the same point this is a point mobility If they are at different points it is a transfer mobility Note that both mobility and impedance are frequency domain quantities

4 requency Response unctions (Rs) Accelerance = Acceleration orce orce Apparent Mass = Acceleration Mobility = elocity orce orce Impedance = elocity Receptance = Displacement orce orce Dynamic Stiffness = Displacement

5 Mobility and Impedance Methods The total response of a set of coupled components can be expressed in terms of the mobility of the individual components In the simplest case each component has two inputs (one at each end) which permit coupling 1 2 General linear system 1 2 The two parameters at each input point are force,, and velocity,.

6 Spring Simple Idealised Elements k f1 2 x1 x2 no mass force passes through it unattenuated Assume f f k x x f k x x f f 1 2 j t e and x f j t Xe Also, block one end so that x 2 0 x So kx K Because X then j j So the impedance of a spring is given by k k k j Note that the force is in quadrature with the velocity. Thus a spring is a reactive element that does not dissipate energy

7 Simple Idealised Elements iscous damper Assume f c f1 2 v1 v2 f c v v f c v v f f 1 2 j t e and v j t e Also, block one end so that v 2 0 no mass or elasticity force passes through it unattenuated f So c So the impedance of a damper is given by c Note that the force is in phase with the velocity. Thus a damper is a resistive element that dissipates energy c c

8 Mass Simple Idealised Elements f1 2 m x f f mx 1 2 f mx f 2 1 rigid force does not pass through it unattenuated f So mx Because X j then jm So the impedance of a mass is given by m m X jm j t Assume f e j t and x Xe Also, set one end to be free so that f2 0 Note that the force is in quadrature with the velocity. Thus a mass is a reactive element that does not dissipate energy

9 Log impedance Impedances of Simple Elements - Summary Spring k k jk Damper c j c Mass m jm Im m c Re m c k k Log frequency

10 Log mobility Mobilities of Simple Elements - Summary Spring Y k j k Damper 1 Y c c Mass Y m 1 jm j m Im Y k Y c Re Y c Y k Y m Log frequency Y m

11 mass spring damper Examples of impedance / mobility infinite beam infinite plate mx kx cx j m mass spring damper jk c Y Y Y mass spring damper j m j k 1 c beam 2(1 ) j ( )( EI) A 1/ 2 1/ 4 3/ 4 2 E 2 plate 8h 2.3c Lh 12(1 2 ) Area A, second moment of area I, thickness h, Young s modulus E, density, Poisson s ratio 2 c E / (1 ) L 11

12 Notes on impedance / mobility Real part of point impedance (or mobility) is always positive (dissipation). Imaginary part can be positive (mass-like) or negative (spring-like). Infinite plate impedance is real and independent of frequency (equivalent to a damper). Beam impedance has a damper part and a mass part, both frequency dependent. Impedance/mobility of a finite structure tends to that of the equivalent infinite structure at high frequency and/or high damping. 12

13 Connecting Simple Elements Adding Elements in Parallel Mass-less Rigid link k c Same velocity Shared force total N j 1 j

14 Connecting Simple Elements Adding Elements in Series k c 1 2 Same force Shared velocity 1 Y1 2 Y2 1 2 Y1 Y2 Y total N Y j 1 j

15 Connecting Simple Elements Adding Elements in Parallel Impedances total N j 1 j Mobilities Y N 1 1 total j 1 Y j Example - SDO System Mass-less Rigid link k m c Note that this representation indicates that one end of the mass is connected to an inertial reference point (=0) m k c

16 Connecting Simple Elements Adding Elements in Parallel m k c m k m k c c Point Impedance k 11 jm c j At low frequency stiffness dominates At resonance damping dominates At high frequency mass dominates Point Mobility 1 Y11 k jm c j j 2 k m jc

17 Log impedance Log mobility requency Response unctions m k m k c c Point Mobility 0.01 Point Impedance Stiffness line 0.1 Mass line Stiffness line 0.1 Mass line Log frequency 0.01 Log frequency

18 Connecting Simple Elements Adding Elements in Series Impedances N 1 1 total j 1 j Mobilities Y total N Y j 1 j Example k c m Point Mobility Y 11 j 1 1 k c jm Point Impedance 1 11 j 1 1 k c jm

19 Log mobility Connecting Simple Elements Adding Elements in Series k c m Point mobility At low frequency mass dominates Y 11 Mass j line 1 1 k c jm Stiffness line At resonance damping dominates At high frequency stiffness dominates 1 c Log frequency

20 Adding a Combination of Parallel and Series Elements Example - SDO System k m c Log impedance m k c total m k c stiffness line mass line Damping line Log frequency

21 E.L.Hixson, Chapter 10 in Shock and ibration Handbook

30 Coupling Together Complex Arbitrary Systems 1 2 General linear system 1 2 An arbitrary system having only two inputs can be represented using the sign convention shown in the figure above ( 1 and 2 both act on the element). When the system is not a simple mass, spring or damper it is necessary to assign both point and transfer mobilities to a system to define completely the inter-relationship between both inputs.

31 Mobility Method 1 2 General linear system 1 2 The equations describing the system are given by Y Y Y Y which can be written in matrix form as or 1 Y11 Y12 1 Y Y v Yf Mobility matrix

32 Mobility Method 1 Y11 Y12 1 Y Y Y Y11 and Y 22 are point mobilities, which relate the velocity at the point of excitation to the force applied. They are defined as Y orce applied at point 1 Point 2 Is free orce applied at point 2 Point 1 Is free Y12 and Y 21 are transfer mobilities, which relate the velocity at the point of some remote point to the force applied. They are defined as Y Y

33 Mobility Method Notes or a linear system Y Y because of reciprocity If the system is symmetric then These mobilities are found by allowing one input to be free The mobility formulation is most useful in experimental work as the point and transfer mobilities are easily measured Re{Y 11 } and Re{Y 22 } must be positive Y Y 11 22

34 Impedance Method 1 2 General linear system 1 2 The equations describing the system are given by which can be written in matrix form as or f v Impedance matrix

35 Impedance Method and 22 are point impedances, which relate the velocity at the point of excitation to the force applied. They are defined as orce applied at point 1 Point 2 Is fixed orce applied at point 2 Point 1 Is fixed 12 and 21 are transfer impedances, which relate the velocity at the point of some remote point to the force applied. They are defined as

36 Impedance Method Notes or a linear system If the system is symmetric then because of reciprocity These impedances are found by fixing all points except one The impedance formulation is most useful for theoretical formulation and when experimentally working on light structures when blocking is possible Re{ 11 } and Re{ 22 } must be positive 11 22

37 Impedance and Mobility matrices for simple elements Spring k k 1 1 j 1 1 Damper 1 1 c c 1 1 Y k is not defined as the element is massless, and the mobilities are infinite if one input is free Y c is not defined as the element is massless, and the mobilities are infinite if one input is free Mass Y m j 1 m m is not defined as a blocked output would prevent motion

38 Coupling together complex arbitrary systems 1 2 I 3 II System I Y Y Y Y System II Y

39 Coupling together complex arbitrary systems - series connection 1 2 I Rigid link 3 II 1 2 When rigidly connected 3 2 System I Y Y Y Y (equilibrium of forces) (continuity of motion) System II Y

40 Coupling together complex arbitrary systems - series connection 1 2 I Rigid link 3 II Combining system equations gives the point and transfer mobilities of the coupled system Point 2 1 ( Y21) Y11 1 Y22 Y33 Transfer 2 YY Y Y The natural frequencies of the coupled system occur when Im Y22 Y33 0 The imaginary components embody the reactive elements which can equal zero

41 Coupling together complex arbitrary systems - parallel coupled system I II or the uncoupled systems When the systems are joined by a rigid link so The point mobility of the coupled system is given by 1 1 1Y 1Y

42 ibration source characterisation Thévenin equivalent system A vibration source connected to a load can be represented by a blocked force b in parallel with an internal impedance i connected to a load impedance l. Source b l Load l l b 1 1 i l i l l i b l Blocked force is the force generated by the source when it is connected to rigid load

43 ibration source characterisation Norton equivalent system A vibration source connected to a load can be represented by the free velocity of the source f in series with an internal impedance i connected to a load impedance l. Source i l Load l f l l f 1 1 l i l f l 1 l i ree velocity is the velocity of the source when the load is disconnected

44 ibration source characterisation Relationship between blocked force, free velocity and internal impedance of the source b = i f b l i l i l f i l i l l f i l l l b i l Thévenin equivalent system Norton equivalent system b b i b i f Similar equations involving mobilities.

45 Reduction of a system to Thévenin and Norton equivalent systems example c 2 c 1 k 1 k 2 b 1 2 b 2 b b m m 3 1 m1 k1 c1 2 k2 c2 3 m 3 Blocked orce 2 b is not included because its ends are attached to two rigid walls

46 Reduction of a system to Thévenin and Norton equivalent systems example c 2 c 1 k 1 k 2 b 1 2 b 2 b b m m 3 ree velocity f b s Source impedance at connection point b Internal impedance s 3 1/ 1 1 1/ 2

47 Reduction of a system to Thévenin and Norton equivalent systems example c 1 k 1 m 1 c 2 k 2 b s b b Blocked orce 2 b 1 2 m 3 1 m1 k1 c1 2 k2 c2 3 m 3 f s Internal impedance b ree velocity f b s s 3 1/ 1 1 1/ 2

48 Summary Introduction to mobility and impedance approach Lumped parameter systems Arbitrary systems Coupling of systems Source characterisation

49 References E.L. Hixson, Shock and ibration Handbook (Chapter 10), edited by C.M. Harris, Third Edition, McGraw Hill. Mechanical Impedance. R.E.D. Bishop and D.C. Johnson, The Mechanics of ibration, Cambridge University Press. L. Cremer, M. Heckl and E.E. Ungar, 1990, Structure-Borne Sound, Springer-erlag, Berlin-Heidelberg-New York. L.L. Beranek and I.L. er, Noise and ibration Control Engineering, John Wiley and Sons..J. ahy and J.G. Walker, 2004, Advanced Applications in Acoustics, Noise and ibration, Spon Press (chapter 9 Mobility and impedance methods in structural dynamics by P. Gardonio and M.J. Brennan).

Introduction to Vibration. Professor Mike Brennan

Introduction to Vibration. Professor Mike Brennan Introduction to Vibration Professor Mie Brennan Introduction to Vibration Nature of vibration of mechanical systems Free and forced vibrations Frequency response functions Fundamentals For free vibration