Chapter 6 Vibration Design

Transcription

1 Chapter 6 Viration Design Acceptale viration levels (ISO) Viration isolation Viration asorers Effects of damping in asorers Optimization Viscoelastic damping treatments Critical Speeds Design for viration suppression 1

2 Acceptale levels of viration Each part or system in a dynamic setting is required to pass viration muster Military and ISO provide a regulation and standards Individual companies provide their own standards Usually stated in terms of amplitude, frequency and duration of test

3 Viration Isolation A major jo of viration engineers is to isolate systems from viration disturances or visa versa. Uses heavily material from Sections.4 on Base Excitation Important class of viration analysis Preventing excitations from passing from a virating ase through its mount into a structure Viration isolation Shocks on your car Satellite launch and operation Disk drives 3

4 Formulate the equation Viration isolator m x(t) k c y(t) Viration source modeled as ase motion 4

5 Formulate the equation (1) F mx ɺɺ k( x y) c( xɺ yɺ ) mx ɺɺ+ c( xɺ - yɺ )+ k( x - y)0 y( t) Y sin t Y : Amplitude : frequency mɺɺ x + cxɺ + kx cy cos t + ky sin t 5

6 Formulate the equation () F cy 0 dr mɺɺ+ x cxɺ + kx F cos t ky sin + 0 t 6

7 Formulate the equation ( Particular solution ) cy cos t X ( 1) () p ky sint X p (1) m ɺɺ+ x cxɺ + kx X p + X p () ɺɺ x + ξxɺ + x ξ Y cos t + Y sin t f0 ξ Y 7

8 Derive the displacement transmissiility(1) X P ( ) + ( ξ ) dr f cos 0 1 ξ dr drt tan dr dr dr X (1) p ξy cos( t ( ) + ( ξ ) φ ) 1 φ 1 tan 1 ξ 8

9 9 Derive the displacement transmissiility() ( ) ( ) ( ) 1 () sin φ ξ + t Y X p ( ) ( ) ( ) 1 1 () (1) cos ) ( ) ( ) ( ) ( φ φ ξ ξ t Y t X t X t X p p p ξ φ tan 1

10 10 Derive the displacement transmissiility(3) ( ) ( ) 1 ) ( r r r Y X ξ ξ ( ) ( ) ( ) r r r Y X R T ξ ξ

11 SDOF under ase excitation x(t) m m k c y(t) ase k( x y ) c( xɺ y) ɺ F - k( x - y) - c( xɺ - yɺ ) mx ɺɺ mx ɺɺ + cxɺ + kx cyɺ + ky (1) 11

12 y( t) Y sin( t) Let SDOF Base Excitation mx ɺɺ + cxɺ + kx cy cos( t) + ky sin( t) harmonic forcing functions The steady-state solution is just the superposition of the two individual particular solutions f 0 c f 0 s ɺɺ x + ζnxɺ + n x ζ ny cos( t) + ny sin( t) x ( t) x ( t) pc ps 1

13 Particular Solutions (sine input) With a sine for the forcing function, ɺɺ x + ζ xɺ + x f t n n 0s sin x A cost + B sint X sin( t φ ) ps s s s s where A B s s ζ f n 0s ( n ) + ζ ( n ) ( n ) + ( ) f 0s ζ n ( ) n 13

14 Particular Solutions (cosine input) With a cosine for the forcing function, we showed ɺɺ x + ζ xɺ + x f t n n 0c cos x A cost + B sint X cos( t φ ) pc c c c c where A B c c ( n ) ( n ) + ζ f n 0c ( n ) + f 0c ζ ( ) ζ n ( ) n 14

15 Magnitude X/Y Magnitude of the full particular solution X p X pc + X ps ( A s + B s )+ ( A c + B ) c f 0c + f 0s ( n ) + ζ n ( ) n Y (ζ) where f 0c ζ n Y and f 0s n Y + n ( ) ( n ) + ζ n if we define r n this ecomes X p Y (ζr) +1 (1 r ) + ( ζr) 15

16 Displacement Transmissiility X/Y (db) ζ Frequency ratio r i Potentially severe amplification at resonance i Attenuation only for r > i If r< transmissiility decreases with damping ratio i If r>>1 then transmissiility increases with damping ratio XpYz/r 16

17 Isolation Effectiveness IE(isolation effectiveness) 1-TR T. R. X Y 1 + ( ξ r ) 1 r + ( ξ r ) 1 ---displacement transmissiility 1. r,tr1, IE0. ( ) 1. 1 r >, r >, TR<1. 17

18 Displacement Transmissiility 3 Transmissiility X/Y ξ0.01 ξ0. 1 ξ0. 3 ξ Amplification region Frequency ratio r Isolation region 18

19 Phase angle ξ0.01 ξ0. 1 ξ0. 3 ξ Frequency ratio r 19

20 Force Transmissiility: F k( x y) + c( xɺ yɺ ) mx ɺɺ T we know that x( t) X cos( t φ), ɺɺ so x- X cos( t φ) FT m X kr X x(t) m k c F T y(t) ase 0

21 Plot of Force Transmissiility F/kY (db) ζ 0.01 ζ 0.1 ζ 0.3 ζ Frequency ratio r 1

22 Isolation in SDOF System Two types: moving ase and fixed ase Three magnitude plots to consider TR transmissiility ratio X Y 1+ (ζr) (1 r ) + (ζr) F T ky r 1+ (ζr) (1 r ) + (ζr) Moving ase displacement Moving ase force F T F (ζr) (1 r ) + (ζr) Fixed ase force

23 Time response Displacement x(t) Displacement transmissiility Time t(s) 3

24 Time response Displacement x(t) Time t(s) Displacement transmissiility 4

25 Time response Displacement x(t) Force transmissiility Time t(s) 5

26 Time response Displacement x(t) Force transmissiility Time t(s) 6

27 SPECIFICATIONS Length: 330m Width: 4m Height aove river: 10.8m Handrail height: 1.m Piers: Concrete and steel Cales: 10mm locked coil Decking: Aluminum Construction: 18m Modifications: 5m 7

28 Millennium Bridge 8

29 Millennium Bridge 9

30 Tuned Mass Dampers Tuned mass dampers to asor vertical movement. Additional tuned mass dampers to restrict the horizontal movement. 30

31 Tuned Mass Dampers 31

32 Tuned Mass Dampers A criss-crossed exoskeleton etween the decking and the cale arms to control swaying. This structure acts as a race. At the centre points of the "x" shaped skeleton to counter lateral movement. 3

33 Viration Suppression 33