VIBRATION STUDIES ON TRIUMF RESONATORS by JIMMY B . A . S c , A UNIVERSITY THESIS THE OF SUBMITTED MASTER OF BRITISH IN REQUIREMENTS LEE COLUMBIA PARTIAL FOR THE APPLIED (1984) FULFILMENT DEGREE OF SCIENCE in FACULTY DEPARTMENT We accept to OF OF thesis required UNIVERSITY OF JIMMY as LEE, ENGINEERING conforming standard BRITISH SEPTEMBER © STUDIES MECHANICAL this the GRADUATE COLUMBIA 1986 1986 OF In presenting requirements BRITISH freely that this for COLUMBIA, available permission scholarly or advanced I agree for understood that gain p a r t i a l degree that reference for by in an extensive purposes Department f i n a n c i a l thesis may or copying or shall not and study. granted her OF allowed MECHANICAL ENGINEERING U N I V E R S I T Y OF B R I T I S H 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: SEPTEMBER 1986 COLUMBIA of by shall I of of the UNIVERSITY make further this the Head this without of It thesis my OF it agree thesis representatives. publication be the Library permission. DEPARTMENT at copying be his the fulfilment for my is for written ABSTRACT The meson Cyclotron f a c i l i t y , resonators the major which when beam structure structural The i n i t i a t e d which p a r t i c l e D i v i s i o n in of a the this vibration to determine v i b r a t i o n . for the types future, Triumf it The 1. the be e n c i r c l i n g vibrating out design funds the vibrating on the nature originating for existing hot-arm is of the from the A hot-arm, would p a r t i c l e on beam. the hot-arm reduce w i l l now at a for a hot-arm be b e n e f i c i a l Triumf. proposed construct a vibration c h a r a c t e r i s t i c s the Kaon kaon In the Factory; producing c y c l o t r o n . modelled cantilever following the s t a b i l i t y to of quality thermal-related studies conducted b e n e f i c i a l the structural of which beam RF experiences desirable national replacement c y c l o t r o n . encompasses experiments requested transversely carried of the design improve s t a b i l i t y the improved w i l l has factory report hot-arm The by Canada's flow-induced spatial of to would resonator, scope replacement place and Triumf, study produced deformation reduces a of a n a l y t i c a l l y beam. to be Investigations a are areas: vibration coolant excitation water flowing forces in the hot-arm; 2. the effectiveness stiffener reducing 3. the and beam of adding damping, to a the lumped mass, cantilever a beam rotational model, in v i b r a t i o n ; minimum-weight v a r i a b l e - c r o s s - s e c t i o n design of sandwich i i a l i g h t l y damped cantilever beam, 4. subjected to a unit t i p , with constraint the minimum-weight harmonic on the subjected 5. a random spectral density the mean square the optimal attenuate 6. to the on influence the tip force the of force beam at the free amplitude; a sandwich rain over design the tip design v a r i a b l e - c r o s s - s e c t i o n point with span, l i g h t l y damped cantilever beam, white with noise power constraint on d e f l e c t i o n ; of a dynamic hot-arm vibration of shape the magnification of the of vibration absorber to and the coolant-flow flow-induced channel excitation in hot-arm. A possible design for the replacement i i i hot-arm is discussed. a Table of Contents ABSTRACT ii LIST OF TABLES vii LIST OF FIGURES ix NOMENCLATURE xiii ACKNOWLEDGEMENT 1. xviii Introduction 1.1 Description of the 1.2 Description of a 1.3 Description Prototyped ) 1.4 1 .5 3. Triumf Triumf of the Hot-arm Cyclotron Existing Hot-arm and Hot-arm 1.3.2 E x i s t i n g Hot-arm with Vibration Absorber Prototyped) Hot-arm 9 Attached Dynamic 12 Hot-arm Vibration 15 on the Accelerating 22 Objectives 23 A Study on the Governing Pipe Conveying F l u i d Equation 2.1 The Motion 2.2 Possible Unstable Hot-arm Conveying Random 3.1 5 9 Existing Effect of Particles 1 Resonator 1.3.1 1.3.3 2. 1 Equation Vibration of of Characterizing 3.1.1 3.1.2 3.1.3 Small Ensemble Processes Motion of a 24 25 O s c i l l a t i o n of Prototyped) Coolant Water Flow the a of Hot-arms Random 29 Vibration Averages 26 and Process 30 Stationary 31 Power S p e c t r a l Density Stationary Process Function for a R e l a t i o n s h i p Between t h e Power Spectral Density and Autocorrelation Function for a Stationary Process iv 31 32 3.2 3.3 3.4 3.5 4. The Nature of the Random Excitation Forces Originating from the Coolant-Water Flow in a Prototyped ) Hot-arm 33 Transversely Vibrating a Vibrating Hot-arm 38 Response Vibrating 4.1 4.2 Study Hot-arm for to a Beam Model of a Transversely Random R a i n F o r c e Measurements on .39 the 45 on the Optimal Structural Which Minimizes V i b r a t i o n . . . . 5 0 The A d d i t i o n of D i s c r e t e Mass-Stiffness-Damping to Reduce the V i b r a t i o n D e f l e c t i o n of a Uniform Cantilever Beam 52 Optimum Design of a Hysteretically Cantilever Beam Subjected to a Unit Point Force at the Free T i p 60 4.2.1 4.2.2 4.2.3 4.3 Calculation Beam S u b j e c t e d Random Vibration Prototyped) Hot-arm An Analytical Geometry for a Cantilever Damped Harmonic Optimality Criterion for the Minimum-weight Sandwich Cantilever Beam E x c i t e d by a U n i t Harmonic Point Force at the Free T i p Minimum-weight sandwich cantilever d e s i g n when the frequency of the harmonic point force i s much l o w e r the fundamental frequency 63 beam unit than ...66 Minimum-weight sandwich cantilever beam d e s i g n when the f r e q u e n c y of the unit harmonic point force i s the fundamental frequency 70 Minimum-Weight Design of A Hysteretically Damped S a n d w i c h C a n t i l e v e r Beam, S u b j e c t e d to A Random R a i n F o r c e with White Noise PSD, with C o n s t r a i n t on t h e T i p D e f l e c t i o n 75 4.3.1 4.3.2 Tip Deflection of H y s t e r e t i c a l l y Damped U n i f o r m Beams S u b j e c t e d t o a Random Rain F o r c e w i t h W h i t e N o i s e PSD 77 Minimum-Weight D e s i g n of a Hysteretically Damped V a r i a b l e - C r o s s - S e c t i o n Cantilever Beam, S u b j e c t e d to a Random R a i n Force with White Noise PSD, with C o n s t r a i n t on t h e MS T i p D e f l e c t i o n 79 v 4.4 4.5 5. The E f f e c t of Adding Prototyped) Hot-arm Discussion Design Vibration 5.1 5.2 on Damping to the Tip of a 83 Possible Replacement Strongback 84 for Various Methods Vibration a Hot-arm to 87 Attenuate the Hot-arm 88 An O p t i m a l Dynamic Vibration Prototyped) Hot-arm Absorber for the 89 5.2.1 Lumped Parameters for a Transversely Vibrating Cantilever Beam E x c i t e d by a Random R a i n F o r c e w i t h W h i t e N o i s e PSD ....91 5.2.2 Optimal Absorber Parameters for a Main System Excited by a Random w i t h W h i t e N o i s e PSD 5.2.3 Damped Vibration Prototyped) Hot-arm 5.2.3.1 5.2.3.2 6. a Weights Turbulence Absorber for Generation in Flow the 93 the 96 Considerations in Selecting Absorber Damping Mechanism The T e s t i n g Absorbers Hot-arm 1D0F Force of on the 98 Damped Vibration a Prototyped) 102 Hot-arm Coolant Water 110 CONCLUSIONS 113 REFERENCES 116 APPENDIX A 121 APPENDIX B 122 APPENDIX C 123 APPENDIX D 126 APPENDIX E 1 27 vi LIST OF TABLES Table 1.1 Page A comparison of the existing and P r o t o t y p e d ) hot-arms. 3.1 Extreme 21 accelerations of the P r o t o t y p e d ) hot-arm t i p . 48 4.1 Vibration for deflection various of combinations mass-stiffness-damping model 4.2 A and damping comparison cantilever beams 4.4 Tip subjected noise PSD. rain on 5.1 Optimum main noise 5.2 MS Test which lumped to uniform beam excitation a sandwich minimum-weight same t i p uniform amplitude. sandwich random design cantilever force the beam rain 72 cantilever force with white 78 Minimum-weight sandwich the cantilever 55 their the for beams of addition, the and with deflection uniform model. between counterparts 4.3 a with t i p a beam, white v a r i a b l e - c r o s s - s e c t i o n subjected noise PSD, to a with random constraint d e f l e c t i o n . absorber system of 83 parameters excited by a for random an undamped force with 1D0F white PSD. 96 results were for the attached damped to the vibration P r o t o t y p e d ) absorbers hot-arm t i p . 103 6.1 A comparison of the hot-arm v i i vibration for the roll-bond panel and pipe-type v i i i panel. 112 LIST OF FIGURES Figure Page 1.1 Plan view 1.2 A 1.3 Partial t y p i c a l tank and Exploded 1.5 Plan 1.6 Partial view of RMS spectrum RMS tank. 2 view of 3 the Triumf Cyclotron an existing roll-bond sectional cutting of 4 the of resonator. hot-arm view plane of is 6 panel. an 8 existing perpendicular to hot-arm) the tip 10 deflection for the hot-arm. main vibration 1.9 a (The length 1DOF of cross existing Cyclotron resonators. view the A Triumf upper-resonator. the hot-arm. 1.8 the elevation 1.4 1.7 of system with absorber spectrum existing 13 of attached damped dynamic system. the hot-arm an tip with 13 deflection an attached for the vibration absorber. 14 1.10 Plan view 1.11 A 1.12 Partial the A A cross (The length of cantilevered upstream 2.2 a P r o t o t y p e d ) hot-arm. 2.1 of end, cantilevered upstream end, P r o t o t y p e d ) hot-arm panel. 16 strongback. sectional cutting the 17 view plane of is a P r o t o t y p e d ) perpendicular to hot-arm) tube conveying vibrating tube 18 in the conveying vibrating in ix the f l u i d , fixed f i r s t mode. f l u i d , fixed second mode. at the 24 at the 28 3.1 Typical a 3.2 power spectra c y l i n d r i c a l Vibration rod. of the Ref. acceleration water-flow noise over [8] of 35 the P r o t o t y p e d ) hot-arm t i p . 3.3 4.1 46 Probability density hot-arm vibration A tip uniform excited 4.2 A cantilever by uniform a rain density 4.3 A 4.4 Optimal S 0 and point beam with an added force with P an beam force 10"*) free mass sin(w added damping white 46 C° noise n M° )t. 53 rotational excited by a spectral 53 cover-plate (T P r o t o t y p e d ) . cantilever A with viscous force the v a r i a b l e - c r o s s - s e c t i o n sandwich static 4.5 harmonic K° of acceleration. beam cantilever stiffenner random function = point thickness excited and force by a for a = 0), (T for beam. an unit 64 undamped harmonic cantilever both beam forces point under at the t i p . 69 v a r i a b l e - c r o s s - s e c t i o n sandwich cantilever beam. 81 4.6 F i n i t e element approximation v a r i a b l e - c r o s s - s e c t i o n 4.7 A replacement 5.1 A transversely attached 5.2 A 1DOF main vibration sandwich strongback damped cantilever cantilever vibration with the beam. design. vibrating system of an absorber. 86 beam with an absorber. attached 81 92 damped dynamic 94 x 5.3 Simplified vibration 5.4 drawing absorber Dependence upon f absorber and 7 the of . 5.5 5.6a 7 5.8b ' +' =0.0l. Optimal Ref. RMS s p e c t r u m of P r o t o t y p e d ) hot-arm to the the the of P r o t o t y p e d ) hot-arm the attached to RMS s p e c t r u m of P r o t o t y p e d ) hot-arm to the the of P r o t o t y p e d ) hot-arm the attached to RMS s p e c t r u m of P r o t o t y p e d ) hot-arm the by values, of . * " * mass For values of exceeds than o p t r stated T r = 7 a a-opt' [40] t i p 101 deflection without t i p for the an absorber. deflection with t i p for an undamped the t i p the absorber for an a d j u s t e d deflection an for undamped the t i p damped 106 the absorber the t i p 107 deflection with (v=0.1 deflection with f o r an adjusted t i p . the damped 0) for an undamped 107 the absorber tip.U=0.15) RMS s p e c t r u m of the P r o t o t y p e d ) hot-arm t i p deflection with xi 06 the tip.(*i=0.05) with 105 1 deflection with 97 the main response less t i p . tip.(<i=0.l0) RMS s p e c t r u m to the hot-arm and 7 the damped tip.U=0.05) RMS s p e c t r u m attached to f contour hot-arm absorber 5.8a a P r o t o t y p e d ) attached 5.7b parameters, of absorber 5.7a attached RMS s p e c t r u m attached 5.6b m f r i c t i o n MS d i s p l a c e m e n t percentage. M=0.1, the t h e MS d i s p l a c e m e n t within optimal of 107 for an adjusted the damped absorber 6.1 Cross sectional roll-bond 6.2 Cross attached pipe-type view hot-arm sectional to of tip.(u=0.l5) the coolant 108 channel in a panel. view hot-arm the of Ill the panel. coolant channel in a 111 xi i NOMENCLATURE mass, of a spring 1D0F mass, of a and main spring damping, respectively, damping, respectively, system and vibration deflection viscous viscous absorber of a transversely vibrating tube or beam spatial time coordinate p a r a l l e l to a tube or a beam a beam coordinate length of a tube or a r i g i d i t y mass per unit length of a mass per unit length of f l u i d mean speed amplitude of of exponential frequency the the resultant tube the imaginary tube or a in beam a tube flow deflection of the argument v i b r a t i o n , parameters beam fluctuation v i b r a t i o n , or beam force f l u i d part define tube the transversely at tube wall-pressure excitation fixed a f l u i d function of turbulent of beam flexural real or of (/) v e l o c i t y vibrating upstream the part by, flow 2 = the from conveying a f l u i d , argument complex argument -1 mathematical expectation probability density x i i tube exit end complex of at i of the function argument (PDF) of the argument variables MS mean square RMS root mean a variance of standard deviation 2 Oy Ry( ) cross value square the value subscript of correlation variable y the variable subscript function with y of dependent variable the y subcript variables in the argument Sy( ) cross spectral variable y density with function dependent of the subscript variables in the argument 0* convection c d^ hydraulic L length 6( ) Dirac speed of turbulence in the f l u i d flow turbulence in the f l u i d flow diameter scale of function the with dependent parameters in the argument s stress in the beam e strain in the beam E e l a s t i c modulus c damping parameter v damping parameter of the of a of beam material viscously a damped h y s t e r e t i c a l l y beam damped beam (') d i f f e r e n t i a t e the argument once the argument twice with respect to t (") d i f f e r e n t i a t e t f ( x , t ) forcing function for xiv a beam with respect to h ( x , T ; a ) unit impulse H(x,u;a) unit complex response function frequency of a response beam function of a beam #j(x),9j(t) j t h normal mode coordinate, vibrating hj(t) similar for Hj(w) a and corresponding r e s p e c t i v e l y , beam to for generalized a transversely (j=1,2,3...) the unit impulse response function frequency response 1DOF s y s t e m similar to function Wj j t h 7j damping the for resonant a unit complex 1DOF s y s t e m frequency ratio for for the a j t h beam normal mode of the beam IJ^(CJ) joint acceptance i/>T transpose of function the j t h f i n i t e element mode shape vector ^ ( x ^ M , mode shape element eigen logarithmic vibration ( )' generalized system non-zero 6 and mass corresponding of to the the f i n i t e smallest value decrement of the damped free response d i f f e r e n t i a t e the argument once the argument twice with respect to x ( )" d i f f e r e n t i a t e with respect to x 0,(x) the f i r s t normal mode #"(x) the f i r s t normal mode xv of of a beam a beam d i f f e r e n t i a t e d twice with respect M° added lumped K°,C° added rotational to x mass stiffener with lumped viscous damping K°,H° added rotational hysteretic a P constant magnitude constant power 0 rain I(x),m(x) much smaller than of moment of spectral inertia per 2B breadth 2D constant planes of unit of the the separation of the thickness, sandwich p, mass d constant .,1 uni' density beam point of force the e l a s t i c sandwich random distance in the the the mid sandwich and mass the cover-plates of the core in thickness of the the beam density, in weight inertia and moment of moment amplitude at t i p of the of the uniform inertia of fundamental d e f l e c t i o n , xv 1 sandwich cover-plates sandwich square of beam between uniform-cross-section weight, and the beam density 3 section beam modulus of cross respectively, cover-plates respectively, WorloiYorNo harmonic length, sandwich respectively, W L the v a r i a b l e - c r o s s - s e c t i o n 2 lumped force mass d ( x ) , E , p with damping A S length stiffener beam in the beams of cross cantilever cross section, beams section, resonance respectively, and of tip mean the W . min reference unit-square weight the of constraint y,N t i p u M tip a / minimum-weight on amplitude MS M m ' a s s r a cantilever sandwich the tip amplitude at the fundamental deflection m uniform t 1 of 0 f the o r t e main beam with resonance cantilever n beam and beams and absorber masses CJ K 2 a a K m m /M /M a m m m f ' m CJ /CJ 'a a C a 7 C 7 m m S^ R(/i,f ,7 , a m /2M a CJ a /2M u m m m m constant power force the on 7 )dimensionless m spectral main MS density of the random main mass mass displacement xvi i of the ACKNOWLEDGEMENT The author generous guidance wishes help during throughout The author assistance to the this is which he had have completed from Mr. Guy Stanford the Cyclotron collaboration. also entire Triumf. of grateful who also D i v i s i o n for assistance the was his of this was his project. His the f i n a n c i a l This professional to thank Dr. project guidance author's useful the for invaluable. Triumf. the wishes Hutton for from without He The S.G. project received not at Dr. course also could supervisor thank immediate K. Fong discussions Triumf's technicians of and is appreciated. The final Engineering of thanks U.B.C. go to and the the Department department's xvi i i of Mechanical technicians. 1. 1.1 DESCRIPTION The OF process injection of cyclotron vacuum feet in of four and twenty the in each two 1.2. Figure sectioned through upper-resonator one another These rows r e s i s t o r s , of (radio alternating Dee plane in radius 1 a l l of the reference resonators, of the 1.1, the view are giant mega-hertz (MeV) With are injected in u n t i l kinetic of enveloped 1 images of to accelerate they have 400 inside plane. of e l e c t r i c a l l y ions approximately brackets a create k i l o - v o l t energy a in tank, c i r c u i t (MHz) 192 the that beam resonating difference path of The the mirror e l e c t r i c a l The with view shows form 23.075 of shown of p a r t i c l e s p i r a l numbers is accelerating of rows plan of tank above 1.1, lower-resonator capacitors axis the two d i r e c t l y Figure and 51.3 lower-resonators. in a the lower-resonators, Figure of of its Inside elevation 1 tank. ground. the cylinder with an [ 1] . mega-electron-volt the height with center shallow 1.3, potential a in the upper-resonator the Gap a located frequency) the near begins t y p i c a l a inductors an beam and to rows resonator RF inches row. A-A about at across A is rows are lower-resonators. ions) e l e c t r i c a l two upper-resonators Figure of and shows acceleration tank v e r t i c a l resonators tank, The 17.05 rows upper-resonators CYCLOTRON (hydrogen tank. oriented are TRIUMF p a r t i c l e ions diameter symmetry there H THE INTRODUCTION (kV) in gained at the KeV of set of the 520 outer energy square Figure 1.1: Plan view of the Triumf Cyclotron tank. 6.5" Figure 1.2: A typical upper-resonator. CYCLOTRON -RESONATOR l-RESONATOR Figure tank 1.3: and the Partial elevation resonators. view of the Triumf Cyclotron 5 gain MHz per orbit, are energy 1250 required beam of turns to at reach p a r t i c l e s is a rotation 520 MeV. used for frequency This of 4.615 resulting sub-atomic and high medical research. 1.2 DESCRIPTION A resonator hot-arm and ground-arm Figure a 1.4. The length vacuum the floor of the at several 123 of the to to hot The is RF s i g n i f i e s R F - c a v i t i e s electro-magnetic allow f i e l d is which two inches which mode the top and a out the inert c a l l e d a is The MHz. l i d or floor the points. the The bottom the from the and panel strongback insufficient Note, voltage is void between the RF-cavity. the word potential RF c y c l o t r o n . l i d of c i r c u l a r ground of The ground has at a approximately inches. formation the is wide. 23.075 panel 123 in Figure at opposite supported shown in a The sub-components, inches RF 1.3. is 31.5 that inside Figure are voltage the ground-arm, shown points; the a are Facing cantilever panel of either e l e c t r i c a l l y hot-arm panel which the which that in distributed tank. panel s i g n i f i e s and is distributed word panel several vacuum shown ground respectively i t s e l f p o t e n t i a l . the at as hot-arm anchored tank hot-arm r i g i d i t y panel is components: consists and wavelength the major a hot-arm panels are is the of hot-arm a the the and three RESONATOR levelling-arm and panel ground has The one-quarter walls TRIUMF ground-arm of ground A consists strongback 1.4. OF at high ground C o l l e c t i v e l y , an The alternating panels are Figure 1.4: Exploded view of an existing resonator. 7 also c a l l e d necessary RF-panels. because cannot maintain period of these of time This anchored to A adjust B inches as The levelling-arm the to but shown The coolant The existing and outlet coolant strongback and the and C. A RF tip water which in of is are as are pipes was the on root end cantilevered levelling-arm of the tank at at point C about point A The distance the strongback strongback consumption currents channel Vibration of the ions three allows and from is is point is 132 B, 123 inches 1.2. quantities 20 of of mechanism pivoting length power panel floor RF-cavity. at i n 3 of / s of the heat. the This manufactured into shown in connected the of the allowed panel to coolant hot-arm end of water was through Both looping ground water inlet strongback. through detected the method the the flows by panels. a 1.5. the removed the 1.0 panels roll-bond Figure coolant flow is through by is plated heat each root to cyclotron copper c i r c u l a t e d incorporated Approximately panel. entire panels coolant-flow hot-arm a or prolonged metallic root l i d the of is supports for the to is panel structural to unsupported water has and of High large B hot-arm clamped hot-arm, Figure total the which size the in A, the the mega-watts. generate the points of of the properties deposition (analogous either movement or to assembly to locations: insulating the of insulated due strongback beam). the e l e c t r i c a l l y their supports. the Cantilevering each when hot-arm the panel. WATER ROOT Figure INLET (OUTLET) END 1.5: PANEL POINT WATER Plan view of a roll-bond TIP FLOW hot-arm ATTACHMENT END panel. co 9 DESCRIPTION 1.3 OF THE EXISTING HOT-ARM AND PROTOTYPE(1) HOT-ARM When its this people called were the improved the this vibration the of pertinent two and to describes hot-arm, the The layers except a Figure section for heat only diagrams model hot-arms deformation leakage) vibration with the the the hot-arm hot-arms, w i l l the be and hot-arm hot-arms. optimal to this shape with Prototyped) The hot-arm A l l the section The of an design existing investigated. prior the vibration. resonators. size in and of investigate given the Triumf, hot-arm Replacement RF existing existing are following the existing attached dynamic hot-arm. HOT-ARM technique. of is structural and replacement with reduce: hot-arms, existing roll-bond of constructed absorber association structural physical EXISTING to minimizes the the vibration form which a hot-arm. w i l l Prototyped) information two report his on needed (source hot-arm 1.3.1 are flow-induced However and work thermal-related hot-arms a at started Prototyped) design the of author alloy the coolant 1.6. The in the hot-arm A roll-bond metal looping panel sheets pattern channel whose irregular roll-bond shape panel is manufactured panel is by fabricated which are bonded which is next from everywhere inflated cross section is of coolant channel the is suspected a to shown to in cross promote Figure 1.6: hot-arm. of the Partial (The hot-arm) cross - cutting sectional plane is view of perpendicular an to existing the length 11 turbulence The related existing extrusion strongback structure. perpendicular to a five half of the strongback. p a r t i a l l y Figure The cut the than hot-arm which the inch-thick a a the reduce the aluminum supported from the by existing of the strongback existing is (see free of tip bending structure stainless aluminum and span hot-arm overhanging two two strongback extrusion the the expense The section shows of the half the panel. webs of aluminum cross strongback, flange at the p a r t i a l I-beam-like to further 1.75 1.6, gravity is of is unsupported hot-arm panel weight is portion steel of tubes extrusion. and the The panel is pounds. The aluminum stainless is in order the forces. inch-wide fact extend combined 280 In excitation of p a r a l l e l under s t i f f n e s s . length 31.5 in deflection Figure the away 1.4) shorter vibration easy steel to see transversely cross shown vibration beam-bending the hot-arm and of with Figure Acceleration hot-arm the the w i l l torsional and root root of this given, like it a non-uniform direction as plate-bending significant of the y r i g i d much with the less more very in are a dimensions beam vibrating The to behave cantilever The vibration. coincides existing 1.4. clamped From hot-arm essentially model B the vibrating Figure of is levelling-arm. that section, in strongback than the cantilever strongback beam at points on a A 1.4. measurements at the test have been f a c i l i t y taken with 20 i n single 3 / s of 12 coolant water Vibrametric the 1030 hot-arm. definition the band RMS is mode RMS RMS deflection frequency, 1.3.2 the is an 0.5*10 ~ EXISTING (RMS) 1.7, transform is given the panel. to the mode of the A of tip with analyzer. a The Section 3. Most at Hz and the damping. The 4.8 of light of the v i c i n i t y tip recorded (FFT) in indication in arm spectrum was integration spectrum hot attached fundamental deflection, is was Figure deflection in the square Fourier response one mean in fast of response narrow root shown 660A through accelerometer The d e f l e c t i o n , Nicholet flowing of area under the of the fundamental inch. 3 HOT-ARM WITH ATTACHED DYNAMIC VIBRATION ABSORBER A l l of dynamic the the existing vibration hot-arm system absorber vibration of hot-arm represented, for in M m , K m viscous K and damping the and C damping C of the m the purpose mode. its system system with of as the hot in mass, arm) M , representation of the not n e c e s s a r i l l y system which The of the absorber are mass is 0.7 and the The Q viscous dashpots actual pound. 1.8. and viscous the a and the in be spring that mechanisms spring the by Figure mass, Note can mode, the absorber. damped attenuating absorber shown (the a Schematically, f i r s t respectively, main f i t t e d attached in respectively, diagramatic weight f i r s t (2DOF) are, are the its motion of are, for the and two-degree-of-freedom The hot-arms are damping viscous. absorber 13 1.4 S - (_> * _i LL. O 'o 1 " 10 FREQUENCY Figure 1.7: existing RMS spectrum of the 15 £0 CHZ) t i p deflection for the hot-arm. MAIN SYSTEM ABSORBER SYSTEM Figure 1.8: A dynamic vibration 1D0F main absorber system system. with an attached damped 14 spring is is a small cantilevered accomplished button on between a of an area t i p under \- s l i d i n g spring. existing hot-arm the dry stainless the cantilever by to the The hot-arm 0.2*10-* RMS plate. f r i c t i o n a l steel plate. surfaces damped is inch as rubbing The the of of RMS Figure damping a ceramic pressure by force another attached indicated curve absorber provided absorber attenuates spectrum The to small the deflection by the of tip the smaller 1.9. </> (_) * Ld Z _l LL. « CL ' o ^ FREQUENCY <HZ> Figure existing 1.9: RMS hot-arm spectrum with an of attached the tip deflection vibration absorber. for the 15 1.3.3 PROTOTYPE(1) The panel. to a Prototyped) The panel 0.032 the vibrating with excitation; as to not sag under as three-layer and this of 1.12. the The r i g i d weight is either side are give the 31.5 with U-turns The run design the along at the change was turbulent-flow the a has shape of roll-bond a riveted the panel are p a r a l l e l as core that shown r i v e t s . in to it thought aluminum aluminum bending be by riveted inch-wide such can together strongback the construction. strongback held e f f i c i e n t soldered pipes strongback four of pipes The I-beams Figure the 1.11 flanges plates. resistance of These to the be more beam. P r o t o t y p e d ) bending in tapered shows The in of roll-bond These in the The is a generation. prestress beam longer coolant 1.10. of strongback gravity. no reduce section sandwich cover-plates to sandwich I-beams sandwich Figure turbulence to length On in cross does the panel d i s c o n t i n u i t i e s used running hot-arm the Propotyped) of sheet. intent was core steel the promote is copper shown Riveting a panel stainless the coolant-channel The has of tip suspected hot-arm inch-thick length made HOT-ARM the upper inch-thick the in strongback the v i c i n i t y in basic v i c i n i t y of discrete the steps construction cover-plate continuous plate of t i p . constructed the The towards of between is the root and to lighter cover-plate the t i p . stations approximately 1 and 7 thickness Figure Prototyped) in 1.11 strongback. 3 is feet in a 0.375 length. WATER INLET (OUTLET) PANEL ATTACHMENT POINT T3T TT" _3_ CP CP ure J3L 3$C T ROOT CD 4 TIP END 1.10: 1 Plan view of a Prototyped) hot-arm panel. LOWER COVER-PLATE VERTICAL 6 I Figure UPPER 1.11: COVER-PLATE A Prototyped) strongback. MM = 1 SCALE: INCH Figure hot-arm. of the 1.12: (The Partial c r o s s - s e c t i o n a l view cutting plane is of perpendicular a to P r o t o t y p e d ) the length hot-arm) i—» 00 19 The lower cover-plate inch-thick continuous 0.375 between The inch upper and inch-thich 4 5, and inches there in lap is stations which has 2 3, and cover-plates plates length neighbouring plate stations lower 0.080 between a 30 but is are no is a machined of down 3 to inches. and Between upper lower 0.625 30 station length. seamed 3 length inch-thick there cover-plates a in and been between inches 0.020 1 4 are station cover-plate 18 cover-plate. together with a The riveted joint. Note station that 1 and the 2 lower accomodate the space is thicker symmetric cover-plate This u t i l i z i n g precious than into the beam which shows not protruding panel. s t i f f n e s s , section is inch-thick hot-arm cross beam more is 1.85 gap. since inches A e f f i c i e n t use cut a in hot-arm order about the core. it of 0.625 groves which coolant should it neutral beam be would be made axis material between The the cannot that the the increase between note of to strongback into pipes done small symmetric has space the about coolant was the of P r o t o t y p e d ) the bending pipes. This built much not here of to encroach that a bending maximize 20 the bending The s t i f f n e s s . consequences construction 1. It are increases strongback 2. It It bending It of cost of tapered cover-plate and under constructing needed a under to counter gravity. e f f i c i e n c y , construction the of requirement structural s t i f f n e s s , increases cost riveting). deflection the the step-wise complexity prestressing increases amount 4. the strongback the the follows. (eg. reduces the 3. as of static with loads respect for a to given material. fundamental frequency of the strongback. The as 1. questions What is Does a the Does a coolant 1.1 RMS water gives hot-arms. which also naturally a minimizes design design for static tip deflection flow structural minimizes strongback strength The optimal strongback frequency 3. arises from the above are follows: strongback 2. which rate comparison loads of of the which tip vibration optimizes vibration structural also minimizes t i p the 3 / s P r o t o t y p e d ) is 0.08*10 existing ? ? the i n the fundamental optimizes the of the which of 20 tip geometry and - vibration hot-arm 3 inch. ? at Table P r o t o t y p e d ) 21 Table 1.1: A comparison of the Existing and P r o t o t y p e d ) Hot-arms. Existing Prototype(1) hot-arm static tip weight of s t i f f n e s s the fundamental damping RMS tip 280 305 frequency (Hz) 4.8 5.05 0.005 deflection that this B C Figure reports is was showed similar pressure between were one vibration that pumping of the in the from at a to hot-arm tank water test the on the pump hot-arms beam frame Triumf's at the at at hot-arms meters and reported mounted c y c l o t r o n . effects f i r s t the vibration the and 0.08*10-3 3 time, Studies in 0.003 measurements Access the those coolant predominantly hot-arm 1.4. smoothing cyclotron, those limited. to the 0.5*10- vibration report hold cyclotron (in) a l l could of 143 (lb) which and 55 hot-arm ratio Note through (lb/in) hot-arm Triumf points inside the the test frame With the large plumbing pipes of hot-arms in not detectable; in the cyclotron near A, engineering were mode a l l 4.8 Hz. the the were 22 1.4 EFFECT OF HOT-ARM VIBRATION ON THE ACCELERATING PARTICLES Vibration fluctuation of variations of the of in the the the RF-cavity voltage resulting ions do path. MeV, the 450 millimeter. vibration p a r t i c l e A Thus as small stable for funds e n c i r c l i n g the existing to 520 MeV synchrotron. The an function important cyclotron now at and at causes the to Dee Because energy, stationary particle to obtain gives subsequent Gap. kinetic s p a t i a l l y of cyclotron the the a the s p i r a l orbits keep of since hot-arm are a upper-resonator is a is the 1.5 hot-arm s p a t i a l l y c y c l o t r o n . in w i l l new hot-arm stable to wall, be be factory from through f i n a l 30 an GeV to serve Factory. on in the the duration Hot-arms the has produced needed of w i l l Triumf routed w i l l vibration shorter it producing a is types future, Factory; and Kaon p a r t i c l e much the the Particle cyclotron the for kaon synchrotron a In Kaon cyclotron anchored tank Triumf. construct v i b r a t i o n . circumferential b e n e f i c i a l proposed existing m i l l i - s e c o n d , of be GeV influence s i g n i f i c a n t period w i l l existing 3 0.27 the advantageous the intermediate is a possible conducted requested The in have is size phase spacing as beam b e n e f i c i a l the it not in beam. experiments be and variation accelerating At hot-arms at center at row ends, attached to the p a r t i c l e cyclotron of time beam only than the the centre of the post. Next to the the hot-arm hot-arm of of an the 23 lower-resonator c a l l e d a d i r e c t l y quadrant), below. located circumferential tank wall, at The copper their socket flow, t i p s . connectors but quadrant the not hot-arms of study to to of would by centre connected to tulip-shaped the plug of one of the matching e l e c t r i c i t y hot-arms of another. hot-arm vibration and and that (also neighbours for enough vibrating post their primarily strong half-row on the a a Thus p a r t i c l e neighbouring quadrant. to to a following single lead the identify identify identify section the the and vibration 4. one a to study resonator Improved an is rather design improved associated of than the c y c l o t r o n . a with whole individual The goals of are: strongback 3. is the independently resonators. identify hot-arm 2. of vibration resonators 1. same effort cyclotron this of n u l l i f i e d the are of OBJECTIVES The the be in vibrate effect between designed connection cannot negative would 1.5 the are Hot-arms nature vibration the which the of the of optimal minimizes optimal a excitation forces on a hot-arm, structural the tip method for geometry of a v i b r a t i o n , damping the hot-arm and the which optimal minimizes shape the of t i p a coolant v i b r a t i o n . channel cross 2. A S T U D Y ON THE GOVERNING EQUATION CONVEYING The the problem problem of conveying f l u i d (1965)[2] have a tube, fixed increased becomes shown at Figure that when upstream a certain and small and the not p o s s i b l e . hot-arm of 2.1. the end OF A PIPE s i m i l a r i t i e s velocity and free and of at the value, amplitude. o s c i l l a t i o n In the o s c i l l a t i o n of a pipe Paidoussis f l u i d perturbations unstable to cantilevered Gregory c r i t i c a l large unstable has vibrating random flow-induced reviewed be in the o s c i l l a t i o n s of vibrating MOTION FLUID transversely shown beyond equation to a a unstable l a t e r a l the of OF flow in other, the is system grow into following, is hot-arm b r i e f l y is shown x 3y(L,t) 9t Figure the 2.1: upstream A cantilevered end, vibrating tube in 24 conveying the f i r s t f l u i d , mode. fixed at 25 2.1 THE EQUATION OF The of a system of studied uniform, r i g i d i t y EI flow line of and mass y(x,t) beam is Gregory motion of and mass The theory x of 3"y(x,t) the m U L, unit with Using at the the the + with a centre c l a s s i c a l that f l u i d the f r i c t i o n , equation of small tube: 3 y(x,t) 2 2 2m U + f (m +m) =0 f 3x3t 2 stream deflection established of a the transverse ox. flexural length 3 y(x,t) f 3x to consists conveying per cantilevered 2 r m, its independent 2 + length having 3 y(x,t) 3x° of coincides and Paidoussis arrived vibration Paidoussis m^ axis tube and is and length perpendicular problem EI unit undeflected bending Gregory cantilever per U. measured dynamical by f l u i d velocity the MOTION tubular incompressible mean SMALL 3t 2 (2.1.1) The boundary y 3y/3x = 3 y/3x conditions = 0 = 3 y/3x 3 2 The f i r s t and the usual s t i f f n e s s vibration problem. respectively, form: forth the terms and of x = 0 at x = L. equation inertia The at forces second centrifugal 2.1 of and and (2.1.2) are, respectively, the c l a s s i c a l beam third terms are, c o r i o l i s forces of the of the f l u i d . Solution A 3 = 0 2 flowing are: Procedure solution procedure is to assume a deflection 26 y(x,t) where = JJe[Y(x)exp(/ ] Re[ arguement. number. co is denotes In The negative. imaginary that zero, the For given a increased second w i l l as the and become the in Gregory Paidoussis. 2.2 and s t i l l UNSTABLE CONVEYING COOLANT is flow velocity its an assumed was checked for the The c o r i o l i s term in for for c r i t i c a l the f l u i d small random large amplitudes. the coolant length of pipe gives water the In the flows hot-arm; the grow case in thus complex part s t a b i l i t y if Paidoussis slowly in more is one or flow OF f i r s t mode. not the have modes. in further The of from velocity increased were is its it unstable studied PROTOTYPE(1) analytical of two tube by HOT-ARM the into net of When this unstable damping vibration directions of the coolant occurs P r o t o t y p e d ) effect of equation p o s i t i v e lateral opposing the of d i f f e r e n t i a l effect the model conveying p o s s i b i l i t y s i t u a t i o n . perturbations a imaginary unstable cantilevered flow is complex FLOW water motion the increased the modes hot-arm, of a w and fourth P r o t o t y p e d ) o s c i l l a t i o n . as become WATER the Gregory OSCILLATION completeness, flow, if w i l l in of frequency unstable higher POSSIBLE For zero. part neutral (m^+m)/m, unstable o s c i l l a t i o n in velocity ratio if is become system real unstable is flow w i l l mass be w the c i r c u l a r system the the system mode also of the w i l l The part shown taking general system (2.1.3) cot)], the the of hot-arm, along the c o r i o l i s 27 force is zero. Inspection system also cannot occur. is that energy tube and the from the shows that A the f l u i d flow. f l u i d flow f l u i d the at velocity forces that U. when from tube gained water vibrating is of inlet must go the be and , and capable the be to vibrating f l u i d velocity in at U. transferred of these is p o s i t i v e . are cannot extract o s c i l l a t i o n fixed is not hot-arms. is of the the inlet external 2.2 to U r , shows of when the can occur, vibrating modes energy tube second mode, the Since f i r s t the Figure this to the outlet, o s c i l l a t i o n outlet P r o t o t y p e d ) two to its than its For in v e l o c i t y from the o s c i l l a t i o n cantilevered when f l u i d . hot-arm the larger energy the a the accumulating only that in of resultant occur, the that shows of unstable vibrate can unstable f l u i d hot-arm unstable existing tube under inlet r to interaction from 2.1 transferred the f l u i d the w i l l this velocity than the During be U for Assume mode, outlet, the resultant f i r s t For must smaller its Figure be can mechanism o s c i l l a t i o n condition system a transfer unstable necessary second modes. in energy vibration conveying vibrating Thus of be energy system. v i b r a t i o n , net energy a hot-arm's coolant the r i g i d from the possible the the for tank, water both a flow. the 28 Figure the 2.2: upstream A cantilevered end, vibrating tube in conveying the second f l u i d , mode. fixed at 3. In a RANDOM V I B R A T I O N previous turbulent f l u i d source random of o s c i l l a t i o n . random in this flow-induced (1984)[8], Bakewell given excited by C y l i n d r i c a l spectral the mode f l u i d by of were c y l i n d r i c a l (1971)[12] motion the rod of and a random of function of the hot-arm on is this a recent of theory l i s t s of Accounts are rods and o n many given by types by fluctuations of Blevins on Corcos induced were i s bodies (1962)[9], rod 29 by i s Chen the to of the the wall-pressure a the be fluctuation. wall-pressure, In and At postulated function rod. under turbulent (1974)[13]. fluctuating vibrates the wall-pressure a vibrating by studies Paidoussis the is (1958,1963,1979)[3,4,5], fluctuating of focus unstable ( 1 968)[11]. turbulent the the the study the c y l i n d r i c a l deflection of as a established studied and C l i n c h of merely vibration well wall-pressure flows of main Crandall fluctuations shapes P r o t o t y p e d ) the problems density correlation to onset random vibration wall-pressure the the of excitation considered the (1984)[7], Vibration outset, is and Nigam (1 9 6 8 ) [ 1 0 ] Wambsganss study random at Accounts Turbulent to tube comparison are (1967)[6] a section, The vibration. references exposed on the perturbation excitation. c l a s s i c a l Lin flow In development section OF T H E HOT-ARMS similar random power cross and manner, a excitation 30 induced by The hot-arm 1. 2. A the study is given. The nature the For as 4. 5. the For beam is span random each t i p . model The ensemble of 1 , 2 , 3 , . . . , n . functions of number with an of n the is hot-arm needed. is modelled beam. c a l c u l a t i o n rain measurements of on force for over a the a P r o t o t y p e d ) random the dependent one necessary e s t a b l i s h But a randomly mounted process for random multivariate usually it is the (t) t, a vibrating for variables of vibrating hot-arm y variable vibration to a signals, independent random PROCESS similar, accelerometer The (PDF). random hot-arm optimal hot-arm response accelerometer the function the forces RANDOM V I B R A T I O N characterize to originating P r o t o t y p e d ) cantilever a terminologies analyzed. n hot-arms to follows. shown. CHARACTERIZING A Consider a vibrating the subjected vibration are in excitation vibrating as P r o t o t y p e d ) forces determine a a vibration excitation to studies, of sections random flow studies, analytical beam The order transversely vibrating the water of vibration smaller random knowledge analytical a of the in flow. random into coolant a hot-arm 3.1 of studied; design 3. water d e f i n i t i o n are from of organized brief are coolant an r = y (t) are r the time. hot-arm probability s u f f i c i e n t is to it To is density establish 31 only the p ( y i f v f i r s t 2 ) r where order and y^ y(t^) = the second and tj order are PDF, p C y ^ discrete and values of time. ENSEMBLE AVERAGES AND STATIONARY Important averages of 3.1.1 an y ensemble. at a Consider fixed mean value: mean square root mean time t a random the ensemble = square value £ [ y (RMS): o If these averages of j ) , the S t i l l function. Consider respectively, process i s at and not a £ [ y i y 3.1.2 2 J ] = said of to important average two of sets two f i x e d y J of 2 p(y)dy (3.1.1) y p(y)dy (3.1.2) (3.1.3) ] - S t y ] times of = £ [ y ( t ) y ( t + r ) ] SPECTRAL i s or = 2 be (3.1.4) 2 t, a (3.1.5) ( i e . independent stationary. the and average just t time i s autocorrelation values y, t the 2 . If yiY2 of function individually, T of and = y 2 , random ° r t 2 the - t, i e . , R(T). DENSITY simply y a r e : 2 ensemble the function, function POWER i s stationary, autocorrelation / = £ [ y 2 independent process another averages or = /variance. are random 2 y ( t , ) across RMS = / M S variance: deviation: = are taken values The ensemble (MS): standard process of E[y] value PROCESSES (3.1.6) FUNCTION FOR A STATIONARY PROCESS Another frequency type domain. of Any ensemble stationary average i s done member of the in the ensemble 32 y(t), represented expanded y(t) y in = s for into the £ C n= 1 time interval function interval, exp(/ n the periodic time large y -s/2 which to is s/2 can identical be with i e . , ncj t) (3.1.7) 0 k. The process a in temporal y(t) is mean given square value of the stationary by: {y (t)J = i sjj^ yMt) dt ! where the member In brackets of the the summation in following form: {y (t)} / = where 3.1.3 Zo, S (a>) f l u i d easily = 0 Aco a n d approaches becomes the power t h i s , the If t In spatial y(x,t) averages, updating the THE FUNCTION wall-pressure flow, by s co an s a single = 27r/Aco. i n f i n i t y integral the of the (3.1.9) BETWEEN considered. s t a t i s t i c a l when co, along spectral density (PSD) y(t). to variables. = o averaging dco, AUTOCORRELATION turbulent nco time 3.1.8 c a l l e d RELATIONSHIP variable Let process is of Prior denotes equation S^(co) function } ensemble. limiting 2 { is POWER FOR (time) many A SPECTRAL STATIONARY was the random on coordinates also random independent are and PSD d e f i n i t i o n s such exposed the space-time autocorrelation previous only bodies the to a independent process, can to AND PROCESS processes fluctuation a DENSITY be its defined include the 33 variable x. Fourier transform c o r r e l a t i o n are, R y ( S y x i » x and cross = / u ) = (1/27T) 2 f T H E NATURE To a l l , pair 2 / f ) FROM the additional s p a t i a l c a l l e d spectral _ t S t OF ( x J 1 , x Tec determine excitation the the density vibration the on To coolant x gives space-time (CSD) a cross functions which measure would Fortunately, it i s p o s s i b l e shedding, excitation the The cross HOT-ARM knowledge necessary; i s i s inner direct form of undoubtedly surface d e l i c a t e to of after the c o r r e l a t i o n complex to ORIGINATING a The many too (3.1.11) the a of the of the pressure c h a r a c t e r i z e . characterize the random q u a l i t a t i v e l y . BLT turbulent separation. i s hot-arm the Boundary-layer-turbulence of design, hot-arm require be FORCES f o r c e s . on CJT) 6T PROTOTYPE(1) a the may sources A (3.1.10) exp(-/ hot-arm the it flow-induced , r ) IN of on and f i e l d 2 6CJ WT) hot-arm the transducers pressure x f FLOW wall-pressure, pipes, l excitation complex fluctuating exp(/ ( x response excitation one. , w ) optimal turbulent-flow BLT the T H E RANDOM E X C I T A T I O N forces of 2 R THE COOLANT-WATER consequence of variable r e s p e c t i v e l y : ( x , , x 3.2 The studied noise are generating following wall-pressure (BLT) i s by some flow bends, empirical fluctuation wall-pressure i s a type researchers. The p u l s a t i o n , c a v i t a t i o n model of proposed by the vortex and CSD f o r Corcos: flow the 34 S(co,x,z) = S(co) A ( c o x / U ) B(coz/U ) C cos(cox/U ), C (3.2.1) C where, x = x , - x and z 2 distances in = the z , - z respectively 2 d i r e c t i o n of flow are the separation and perpendicular to flow. cu i s U the i s C S(co) frequency the i s speed the of at the which PSD wall-pressure the BLT i s function of f l u c t u a t i o n . convected. the BLT wall-pressure f l u c t u a t i o n . A(CJX/U ) and c c o r r e l a t i o n B(coz/U ) amplitudes perpendicular In to this empirical i s independent CSD separation For low c y l i n d r i c a l 1 ) model = of for the S(co,x frequency r o d , Chen A(wx/U ) = e x p ( - 0 . 1 |cox/U B(coz/U ) C where U i s = free Typical are shown in the surface thickness stream power Figure of a of cross flow and homogeneous turbulence, the locations and dependent , z S(co). on the 2 2 , z 2 ) = p a r a l l e l (3.2.2) water flow over a suggest: e x p [ - 2 . 2 ( c o » c / U )] C | ) C the a and exp(-0.55|coz/U | K i s the 0.4 d i r e c t i o n a n d Wambsganss + /U , x 2 0.6 c the the i e . , = U in are flow. distances, S (co, x , , x , , z , , z respectively C 1 ) , of (3.2.3) the turbulent boundary layer and v e l o c i t y . spectra measured 3.1 [8]. inch rod The in a by Chen spectra 2 inch and Wambsganss were measured on water channel. In 35 Figure over a 3.1: Typical c y l i n d r i c a l rod. power spectra of the water-flow noise 36 the Strouhal are nearly approximate S(«) = Q 2 ( p w number range constant and constant PSD: U 2 ) d 2 10 > tod^/2ir\J Blevins > 0.1, suggests the spectra the following , 3 (3.2.4) where Q2 _ 9.4*10" is the s e c / f t 7 hydraulic Approximate 3 , p is w the density of the in a water and d^ diameter. Length Scale of the BLT Prototyped) Hot-arm The Reynold's number of the Prototyped) hot-arm panel at evaluated in Appendix A, given Reynold's No. The = Strouhal evaluated given the f No. is hydraulic = the of s l i g h t l y lower length scale hot-arm of the fd h /U the of since come from = / s of coolant a water, by: BLT wall-pressure frequency 5.05(0.31)/40 fundamental coolant Wambganss. would of fundamental diameter velocity and 3 in of fluctuation the hot-arm is by: Strouhal where i n flow 1.1*10*. number at is 20 turbulent than But the most the fundamental frequency (0.31 in), water the the BLT (40 and i n / s ) . experimental most at important the of the frequency. 0.04, (5.05 U is This to Hz), the is the hot-arm in the the stream number frequency wall-pressure is free studied number the d^ Strouhal cases fundamental contribution PSD = by is Chen integral of the vibration v i c i n i t y 37 The percent convection of distance is U. for The the are length the as scale is energy L « The hot-arm 0.6U/w which form one rain S ( w , x , , x f where 6 is may ) = a be for Summary on Prototype(1) The excitation is case a 2 the approximate zero function is integral length components since frequency. The 0.8 in the « L. length is of not correlation white noise points d i r e c t i o n , in the the very length, with no space. CSD of For this as: ) , function. (3.2.5) For d i s t r i b u t i o n a n a l y t i c a l Nature correlation situation between flow written of 0.37, by: than spatial excitation in This = or the the given 60 of the just an the long beam-like of excitation hot-arm forces is studies. Flow-induced Excitation in a Hot-arm above on r a i n , further the to smaller inches. S(CJ) 6 ( x , - x Dirac scale limiting case dimensional adequate A 2 123 the dimensional random one of frequency proportional much The to separation 1/e c o r r e l a t i o n distance. higher is fraction functions, cross equal the Since 0.6(40)/(2)TT(5.05) random correlation a c for length scale the to U /o>. decay is = is from of to the integral length different a for d i s s i p a t i o n « distance, attenuate L, eddy smaller U / u approximately decay equal scale, the approximate to is c exponential-decay integral same u" eddy CSD approximately amplitudes speed the was hot-arm is example. l i k e l y to be The more flow-induced complex than 38 the BLT excitation. However, flow-induced excitation total length of f l u i d and stream. thus The Wambganss content took data that the lower the for hot-arm's PSD a the would density flow-induced be VIBRATING is section, has a it and random a Clinch, the a BLT in and frequency none with of of them v i c i n i t y to varying model down wall-pressure the force the Chen reasonable slowly to viscous wide-band is rain a the quickly Although possible on is very frequencies of assume in those white noise the actual hot-arm. CANTILEVER studies, vibrating of a hot-arm cantilever coolant material dimensional of of comparison water decay PSD scale BEAM MODEL OF A with a HOT-ARM conveying structural a the range. VIBRATING analytical transversely flow wide-band Thus in Bakewell, frequency, excitation TRANSVERSELY would the low length small since frequency very be by that water low frequencies. For taken fundamental spectral 3.3 hot-arm indicate of in should disturbances data a l l fluctuation the the water the s t r e s s - s t r a i n beam beam is with within is relation modelled a its the cross structure. postulated of uniform to obey The a v i s c o e l a s t i c one type, i e . , S = E(e where s + ce), and modulus of and dot the e (3.3. are, respectively, e l a s t i c i t y denotes and c is stress the differentiation and viscous with strain, damping respect E is 1) the constant, to time. 39 The bending moment of 3 y(x,t) = + Modifying damped ). undamped of motion is + m,U 2 derivatives. = 3t because of c o r i o l i s force, terms of centrifugal comparison to usual the 3.4 c l a s s i c a l RESPONSE a excitation to force, at c a n be i e . , = no force, obtained and t t by 0 For a i s and to are with i t s normal force third Triumf taken mode and hot-arm, be small in neglected; the or without difference. shapes the Now and the their employed. A TRANSVERSELY VIBRATING BEAM FORCE the response superposition excitations. = second, are y c l a s s i c a l the detectable c a n be of centrifugal and they beam structure, unit a possess hot-arm, T O A RANDOM R A I N linear x term CALCULATION FOR standardized applied not terms a undamped conditions SUBJECTED For c o r i o l i s showed (3.3.3) independent (3.3.3). of 3x3t f ( x , t ) , is damping stiffness flow, orthogonality the f £ respectively, frequency water does equation and the fundamental coolant the 2m U 2 2 function system 2 y(x,t) (m,+m) This the 3 y(x,t) + 3x 2.1, by: 2 1 forcing equation 2 3x"3t a motion, 3 y(x,t) cEI f l f ( x , t ) , forth given 5 t and of 3 y(x,t) + modes (3.3.2) equation 3 where by: 2 equation + given 3x 3t 3"y(x,t) 3x c 2 the EI i s 3 -EI( 3x beam 3 y(x,t) 2 M the a A unit standard to of a general the response impulse response form of excitation 40 f(x,t) The = unit y(x,t) the r t - t unit The at = 0 x . = = Since exp(z unit exp(/ 6(x-a) transform of h(x,co;a). R denoted as of at a the excitation fixed i s frequency co (3.4.3) frequency response function is equation it space-time (3.4.4) cot). in (3.4.1), as be a s : term by form force complex of obtained to cot). transform The i s i e . , H(x,co;a) the standard harmonic a, state denoted y(x,t) function (3.4.2) Another 6(x-a) steady be (3.4.1) response complex applied ) . 0 h ( x , r ; a ) , = f(x,t) 6 ( t - t impulse = where to 6(x-a) follows cross (3.4.3) that superposition = Z of the the Fourier i s Fourier H(x,co;a) correlation the is of the unit response impulse is responses follows: y ( x 1 r x 2 , T ) / a cU>i J Z M x , , * ? , ; * , ) where 0, and 2 coordinate, spatial the cross S (x y 1 f and the £ 2 coordinate correlation of are 8 of the response x ,co) 2 = /Q as d £ , 2 , 0 2 ; £ integration are the and R^ obtained R U, d £ , 2 h ( x excitation i s correlation ad 0 2 F ) d £ 2 the force. by S ,T+e,-d ) 2 2 , (3.4.5) variables integration i s r for the variable time for space-time the cross The c o r r e s p o n d i n g Fourier transforming CSD the follows: /jj H ( x , , - c o ; £ , ) H ( x 2 ,co; £ 2 )S f U, ,£ ,co) 2 d £ (3.4.6) where Sr i s the CSD of the excitation. 2 41 The can be impulse found h(x,t;a) = in Z #j(x) terms g.(t;a) j where response of of i t s the jth normal normalized that such as dx k their unit and length integrate + The ccj2g g k (t;a) = impulse response for h(x,t;a) = H 1 k (w) j * k, the <6j(x) be generalized = are the the x from of the = 1,2...n) length a n d mass equation resulting 0 to normal L (3.4.7) and modes apply [14] to equation same beam as (3.4.9) the with zero one-degree-of-freedom function. i s given R Thus the unit the = mL[toJ-£J 2 i n i t i a l as given by: + / 3.4.11 gives: (3.4.12) the 27 w co], k k impulse (3.4.11) R same (1DOF) by: k R function y i e l d : (3.4.9) # (x)0 (a)H (w), is the (3.4.10) response R by by: the the per and equation k k Z k masses (3.4.8) Substitute Z 0 ( x ) 0 ( a ) h ( t ) . k transform of equation H (u>) response Let i s 0 (a)5(t)/mL. given i s k k and gj(t) k unit where = k k h (t) H(x,w;a) (3.4.7) h (t)c6 (a), where Fourier over to is shape (k multiply conditions solution conditions beam. (3.3.3), + o,2g k k the tf> (x), k for = of into g 0 j respectively, orthogonality mode corresponding m, (3.4.1) k 1,2...n), coordinate. = mL f o r = L = i e . , follows: m^j(x)tf> (x) where modes, (j generalized /Q normal beam 3 i s defined uniform-cross-section 4>.{x) 1 corresponding are a 1DOF u n i t complex frequency (3.4.13) 42 and t h e damping 27 = ecu .. 7 i s given by: k (3.4.14) 2 k Substituting S ( x , ,x ,o>) 2 equation = Z Z y where, I ratio j k ( w The ) j ^ function into (3.4.6) 0 • ( x , )<t>, ( x ) H . ( - w ) H . J K k (« )S 2 Ij (<j) i s k f U, , * y i e l d s : .. (u) 2 i 'o^O * j U i ) 0 = (3.4.12) (3.4.15) K _ J K 2 ,") known d £ , d £ as the . 2 (3.4.16) joint acceptance function. Vibration Response The to a joint k = used D X F O by a random i s evaluated R :5= f o r rain of equation a to be: K (3.4.8) (3.4.17) the integral in equation to mL/m. non-uniform to evaluate finite a forcing f o r j * k . i s equal For Force function excited (3.2.5) 0 virtue 3.4.17 be equation beam = S(w) JQ Ij (w) By of Rain acceptance uniform-cross-section function Random beam, the element the finite reponse. Consider discretized uniform-cross-section beam element method the can following system f o r a given b y : [M]v + [ K ] v = 0 , where v , [M] a n d coordinate denote inner of vector, [K] a r e , mass the normalized product equation i//j[M]\J>j (3.4.8). respectively, matrix finite and stiffness element i s analogous Usually the mode displacement matrix. shapes. Let Then to the generalized the finite element the mass generalized 43 mass is normalized element case, to the equal analogy of acceptance function of The reason for existence the individual matrix are (3.4.8). with the terms the dissimilar product" ^j[M]\//j (3.4.17), the system are set of the from sectional [M] is a when the mass matrix per with well resonance peaks into (3.4.15) =* S ( u ) L For a random S , integrating residue Z c6 ( rain of the equation beam elements the of in mass a l l "inner equation matrix the of elements mode damped structure substituting (3.4.8) equation gives: (3.4.18) 2 k u yields the a noise range following PSD, where > S(w) - « > MS deflection: <J °° b y = the 2 Z — k . (3.4.19) CJ£ approximation is summation. s o l u t i o n mode approximation f i r s t applying on <6 (x) intuitive the the lightly the [4] mode a over * one for S (x,w) also of to discussion white 2 a to length with cm L Often mass |H ( ) | . x ) y theorem ( x , t ) ] equal force 0 £ [ y element integral separated, and 2 S ir 2 that properties, the with 0 i s , similar the unit 1/m. unity. beam, y to joint to analogy many finite the equal finite composed of analogous uniform-cross-section S (x,u) integrals the in is simple consistent cross Continuing (3.4.17) a is where to of for integral (3.4.17) system is Then the equation evaluated When unity. that most would resonant of come is the from frequency, adequate; of the the order contribution the to,. S(CJ) in Thus for convergence 1A>£« to the the S(co) It is first v i c i n i t y slowly 44 varying OJ, , near deflection is given S(tO, £ [ y 2 ( x , t ) ] the one mode approximation of the MS by: )7TC/> (x) 2 » . (3.4.20) m Lca>i 2 The energy vibratory when the d i s s i p a t e d system increases amplitude of e l a s t i c bodies, energy d i s s i p a t i o n frequency, [14], For is the a £ [ y v is The given per a viscously frequency of is constant. But damping is the MS damped vibration for than viscous deflection the f i n i t e L whose of is the damping given by: (x) , 2 many mechanism, independent approximation damping, 2 by the cycle * m where with hysteretic hysteretic ( x , t ) ] cycle vibration better Stw, )ir0 2 per (3.4.21) vco 3 hysteretic element damping constant. equivalence of equation 3.4.21 is by: S f w , ) * U T [ M ] t f , )H ( x ) £ [ y 2 ( x , t ) ] « , ?M w 2 where M, u s a l l y is a vibrating w i l l modal normalized Now force the over be design. one beam the needed to mode later 3 mass in the fundamental mode which is unity. vibration subjected beam (3.4.22) to a response one span has been in the study of a dimensional e s t a b l i s h e d . for the transversely random These optimal rain results hot-arm 45 3.5 RANDOM VIBRATION MEASUREMENTS ON THE t i p was PROTOTYPE(1) HOT-ARM Random the vibration following signal spectra and Krohn-Hite analyzer. showed higher that beam transducer signal at and 5.05 The Hz f i l t e r i n g acceleration The in damped 1DOF average equal system The the expected of are response the the are small. envelope were analysis analyzer. tip small, The deflection their tip large. the tip typical by for any a second mode the 660A deflection signal although a Gaussian of zero a of the l i g h t l y crossings Because two a noise or of the in a whose hot-arm envelope random is beat harmonics beating the system narrow-band varying y(t) white narrow-band similar slowly acceleration response frequency. shows f i r s t Nicholet together, figure for modes with close hot-arm frequency fundamental frequencies of the vibration associated amplitude the of a Nicholet from the excited phenomenon in is and contributions were plot p i e z o e l e c t r i c tip higher 3.2 of with the because contributions frequency to the measured 1030 f i l t e r f i l t e r e d into needed Figure excitation. of band domain band Inspection channeled time plotted Vibrametric vibration was from hot-arm 3750 modes was contributions the apparatus: accelerometer, FFT of with fashion [15]. The a normal flow PDF or of y, p(y), Gaussian turbulence is a plotted in Figure 3.3 approximates d i s t r i b u t i o n . This is process many independent with plausible since random 46 cu Ld O U TIME ( S E C ) Figure 3.2: hot-arm Vibration acceleration of the P r o t o t y p e d ) t i p . 5.66 : >> CL 0.0 -0.854 -r 0 0.854 T I P A C C E L E R A T I O N , y CIN/S > Figure hot-arm 3.3: t i p Probability vibration density function a c c e l e r a t i o n . of the P r o t o t y p e d ) 47 sources A and Gaussian in a such input Gaussian following a random linear output. A single-variable of vibratory system w i l l Gaussian PDF }. o~ to and standard 0.08*10~ 3 Figure 3.3, the mean deviation crossings. 0.078 where Note of of deviation variance is it is would equal be bigger acceleration. one extreme interval. that A set value The y denoted y by denoted y by the than of by is mean to value. at the time extreme extreme sampled equal RMS standard data for is every a» of is plot the zero a.. y is of the / w of 2 = zero the and the of the y, acceleration deviation shown is zero, value domain values to frequency value the MS the deviation expected the equal look 2 standard the since the is . is is another evident to, to 2 denoted that standard Taking i n / s deflection i n , the (3.5.1) 1 t i p has 2aj According hot-arm result 2 exp{ approximately Gaussian. d i s t r i b u t i o n : - ( y - y ) = the approximately a form • (2*) is into 1 p(y) process in twenty of the Table 3.1; second time 48 Table 3.1; t i p . ( i n / s 2 Extreme accelerations of the P r o t o t y p e d ) hot-arm ) 0.20 0.12 0.15 0.13 0.74 0.13 0.11 0.20 0.13 0.15 0.092 0.074 0.15 0.13 0.12 0.092 0.11 0.092 0.12 0.15 0. 0.11 0.092 0.15 0.12 0.13 0.13 0.12 0.13 0.092 0.18 0.17 0.18 0.13 0.18 0.16 0.21 0.18 0.11 0.15 0.12 0.15 0.061 0.13 0.12 0.074 0.18 0.12 0.13 0.13 12 Mean-extreme This times good shows the estimate, vibration Estimate The to on be of in the a a the y =* note a one as 3a». e theory 2 usual of by Excitation as i n / s deviation P r o t o t y p e d ) to 0.13 g i e . , the modelled subjected y = that standard references to value: practice the The hot-arm on dimensional three value find with is a further application [16]. a excited transversely can values Davenport Force taking mean-extreme reader extreme of P r o t o t y p e d ) by the vibrating random rain Hot-arm coolant flow cantilever force over is beam the 49 beam span. As mentioned this cantilever 1.4. Vibration beam T h e beam PSD random from this the f i r s t measured mode function f i n i t e a of B. that material rain MS with the force section, points f i r s t mode at o) , of Stw,), the root alone h y s t e r e t i c a l l y y the A and B of is deflection method. of Figure i s to damped. c a n be hot-arm material. be The estimated t i p attributed fact that model actual = in the the the riveted hot-arm has a i s t i p has a o n e mode are in the parameters of model i s same for to of as that of model i s for aluminum moduli be a f i n i t e can be and the cantilever l b / i n where of 140 element in selected P r o t y p e d ) s t i f f n e s s the shown of 109 the modelled the p s i e l a s t i c approximated t i p as 6 with i s beam the 10.6*10 construction measured hot-arm is the s t i f f n e s s evaluated modulus usual acceptance beam. as the l b / i n . model are by: 1 i/>,(L) a frequency discrepancy hot-arm The given to of The e l a s t i c than joint cantilever modulus fundamental This and hot-arm element e l a s t i c hot-arm. mass The P r o t o t y p e d ) f i n i t e p s i , lower 6 modal Prototyped) The i t s actual 6.5*10 M, the non-uniform such The in shape, element Appendix the previous mode. The as a coincides response considered. of in (*![M]*i) l b * i n , = 2.95 i n , = 31.7 For variance of the t i p the strength of the excitation (3.4.22) to be: Siujv/v = 255 = 8. 9* 1 0 " 3 / s 2 , rad/s d e f l e c t i o n : i s i n o* = 6.0*10 approximated 8 l b * i n . - 9 by i n 2 , equation 4. AN A N A L Y T I C A L STUDY FOR An A study structural reduce hot-arm by transversely a a n a l y t i c a l is to determine what stiffener, response a the or It was shown dimensional random rain possible model hot-arm. The harmonic point the tip free The addition of the other, a force of the result study structural of better like at model of minimum-weight either the the is design unit hot-arm, than harmonic is made the beam that noise a PSD excitation frequency is a one is a on a unit applied find the to i e . , beam. a continuous continuously For an hysteretic viscous optimal e l a s t i c damping damping. h y s t e r e t i c a l l y cantilever point 50 of involving the a sandwich vibration excitation, to material, of the mass, mass-stiffness-damping hot-arm cantilever damping for beam. extended a discrete section fundamental structural the of model investigation models white would replaced comparison random type then Triumf A two with discrete of of on previous the v a r i a b l e - c r o s s - s e c t i o n to force cantilever of the a for flow-induced v a r i a b l e - c r o s s - s e c t i o n structure in this has beam. is beam of the which structure adding damping standard geometry d i s t r i b u t i o n that determine hot-arm point responses e x c i t a t i o n . to cantilever cantilever beam a hot-arm starting effect uniform analytical The for GEOMETRY VIBRATION conducted vibrating The STRUCTURAL MINIMIZES geometry rotational of OPTIMAL is vibration. studies. of THE H O T - A R M WHICH analytical desirable ON force or beam, the is a Thus damped subjected random rain 51 force, with constraint investigated. designs is material: Size in and to The to make put the r e s t r i c t i o n the form beam reported analyses. of a depth. on rational on on for the the hot-arm use where is after possible a careful of it w i l l replacement construction do onto between most the the hot-arm consideration is minimum-weight the imposed dimensionless ratio A v i b r a t i o n , investigating e f f i c i e n t material t i p of beam good. designs length design the is above 52 4.1 THE ADDITION REDUCE THE OF DISCRETE MASS-STIFFNESS-DAMPING VIBRATION D E F L E C T I O N OF models the A UNIFORM TO CANTILEVER BEAM Let models A1 A three ' A T , uniform fixed A2 A at A labelled B1 A A as C1 as the Let two 'C1' light the shown A light into of the at end added to model shown mass be labelled unit length as per m is x=L. in and A1 at x=a as shown in of 4.1, with the analytical and force modified span in is attached to model A1 Figure 4.2, where of beam P sin(oj t) A << L. excitation be 'B2'. Figure the K° models point force beam of where beam white as models and co is n P noise shown of is n the could PSD, in applied fundamental be S 0 , Figure damping be at unity. is applied 4.2. labelled as ' C 2 ' . viscous via damping complex damping beam parameter modulus parameter hysteretic hysteretic K°. as rain beam viscous C2 free stiffener 'B1' harmonic over A with analytical models random models x=a+A two frequency B2 is rotational and unit x=L, beam and M° beam 4.1. x=a Let x=0 cantilever ' A 3 ' . cantilever mass lumped at and end lumped Figure A3 'A2' of via damping C° damping complex is added parameter H° is introduced approach modulus parameter c is in and p a r a l l e l v is approach added a in into lumped to K°. introduced and a lumped p a r a l l e l to 53 Figure 4.1: A excited by harmonic Figure 4.2: rotational random a rain uniform A point uniform stiffenner force cantilever with K° force beam P sin(a> cantilever and white viscous noise with an mass M° )t. beam damping spectral added with C° an added excited density S 0 . by a 54 For M ° « 2 ( a ) small « mL, H A #" (a) « c v 0 2 << 2 1 and accuracy in i s reference rate of method the can fundamental simple the mass-stiffness-damping K°A 07 (a) 2 2 vco mL) 1) a needed, [17], mode the vibration more model. but analysis to ^ 2 ( a ) in i s « the beams method, T h e mode does are a for beam lend o n e mode the of itself analysis combinations excitation tabulated in more acceleration the shows If convergence approximation C ( i e . , described the not and 2 accurate method ( i e . , cco mL, adequate. increase procedure. addition, results 2 acceleration deflection The C ° A damping used Appendix mass-stiffness-damping damping mode c a n be frequency solutions. 2 o n e mode summation give <j mL, and l i g h t 2 « « addition Table model 4.1. the to of of and 55 Table 4.1; for Vibration various deflection Model y(x) B1; - Model addition, uniform of beam cantilever the excitation P K°=0, sin(cj t) n C°=0, model H°=0 Model CI: viscous damping Model C2: hysteretic Model CI: viscous Model C2: hysteretic P0,(L)«,(x) • mLcw i B1 : P sin(a> t) n damping Pt6, ( L ) * , ( x ) y(x) m L vix>\ Model B2: S 0 S ?r0i o £ [ y 2 damping (x) ( x , t ) ] m Lcco^ 2 Model B2: S 0 S JT<6 0 £ [ y 2 ( x , t ) ] 2 (x) * m Lfwi 2 beam lumped model A1: M°=0, Model a combinations mass-stiffness-damping damping of damping and continue Model table A2: Model . . . M°#0, B1: P K°=0, C°=0, sin(a> t) H°=0 Model n C1 : viscous damping P*, ( L ) * , (x)^{l + (M°/mL)<6 (a)} 2 y(x) mLco) 3 Model B1: P sin(a> t) n Model C2: hysteretic Model C1: viscous Model C2: hysteretic damping P<4, ( L ) ^ , ( x ) y(x) m L I>CJ Model B2: S 2 0 S TTC4 0 £ [ y 2 ( x , t ) ] 2 damping (x) * m Lcw? 2 Model B2: S 0 S 7T0 O £ [ y 2 2 (x) ( x , t ) ] m L v c o V ( 1+ ( M ° / m L ) 0 2 2 (a)} damping 57 continue Model table . . . A3: M°=0, Model B1 : P K°*0, C°#0 sin(co t) or H°#0 Model n C1 : viscous damping Ptf, ( U t f , ( x ) y(x) m L { c w + ( C ° A / m L ) c 4 7 ( a ) }v/{cj + ( K ° A 2 Model B1 : P 2 2 sin(a> t) 2 Model n 2 / m L ) 0 V C2: hysteretic 2 ( a ) } damping P0, ( L ) $ , ( x ) y(x) mL{^cj Model B2: S 2 + (H°A /mL)0V (a)} 2 2 Model 0 C1: viscous S 7T0 2 O £ [ y 2 2 Model B2: S 2 0 2 Model 0 ( a ) }{u> + ( K ° A / m L ) 0',' 2 2 O ( x , t ) ] 2 C2: hysteretic S 7T0 £ [ y (x) ( x , t ) ] m L{cu> + (C A /mL)<f>'J 2 damping 2 (a)} 2 damping (x) m L{^co 2 2 + (H A /mL)0V o 2 (a) }/{cj 2 2 +(K°A /mL)c6^ 2 2 (a)} where, a>, : the 0 (x) : 2 M°=K°=0; of x mode frequency the of it square i s positive the of the f i r s t beam normal and increases with for M°=K° = 0. mode shape increasing for value [17]. <67 (x) 2 fundamental : the shape for square of M°=K°=0; the it curvature i s of positive the first and decreases normal with 58 increasing The value of following model x [17], observations A2-B1-C1 can be M° increases shows that and A2-B2-C1 made: the resonant amplitude. models A2-B1-C2 effect on model A2-B2-C2 the resonant and shows show MS that M° has l i t t l e deflection. that M° attenuates the MS deflection. models that A3-B1-C1, C° model and models Added deflection A3-B1-C1, the in deflection and damping. a l l H° For models at x=0 (extension is of of H° MS and A3-B2-C2 and MS l i t t l e show deflection, effect on elastic of the is the on the that K° and MS the effects model of of that suggests resonant independent and effective discretely the the like amplitudes. This deflection. damping M° show structure resonant that A3-B2-C2 attenuate However indictates most x=A. and which the 2 and an shows 0" (x) H° has and and dependent model attenuates damping C° damping, better function resonant K° A3-B2-C1 cases. are Inspection damped the that resonant dampings the hysteretic shows A3-B2-C1 amplitude. attenuate the attenuate A3-B1-C2 resonant is H° A3-B1-C2, K° have The when that modified) M° and beam on excitation the the K° hot-arm, frequency, [13]. amplitudes a of of hysteretically l i t t l e effect characteristic rotational its the while of stiffener attachment points continuously minimum-weight the with are modified cantilever 59 beam be design one i e . , with which a beam is on the stiffened in bending which of tapering, i e . , is not at known models d e f l e c t i o n . is tapered l i n e a r l y , this Inspection damped constraint of shows The K° effects regions. At frequencies is damping The regions that of attenuation. 0" (x) 2 the MS the most cantilever that M° when be (extension beam one has This of design which attached is of in the the MS three resonance, force) from inertia is is the force) resonance the dominant. a l l three with the fact independent of the a l l give K° and H° functions at and x=L, x=A that on a the tf> (x) 2 and to K° and with attenuate continuously modified) constraint MS from the x=0 the into away force) effective at e t c . , frequency the near M°, suggests heavier divided and is discretely with The combined why on s t i f f n e s s and force most H° contributions explains is attenuate from damping c h a r a c t e r i s t i c s d e f l e c t i o n . modified should The on geometry h y s t e r e t i c a l l y generalized response, damping vibration, away above (or generalized deflection effective MS and end, l a t e r . of be fixed The follows. can should c u b i c a l l y and generalized force frequency dictate system frequencies (or t i p . a l l K° as frequencies hysteretic frequency H° At that of at H° M°, below (or inertia force fact 1DOF force dominant. of and explained damped the discussed M°, a the is that of resonance it deflections be while the MS response dominant near the can s t i f f n e s s towards amplitude p a r a b o l i c a l l y , point? deflection the resonant minimum-weight MS v i c i n i t y deflection of the free 60 tip and geometry is 4.2 stiffened of this in l a t e r . OPTIMUM DESIGN SUBJECTED FREE OF A TO A fixed end. minimum-weight DAMPED HARMONIC and A POINT Buckling of The beam CANTILEVER FORCE and AT THE Shape found of d i s t r i b u t i o n of c o l l e c t i o n of in plates, fundamental a two Distributed type as and of volume Parameter l i t e r a t u r e is follows. shells frequency of bars, beams, and compliance of structures response of two and Multi-purpose category sub-categories, forced be design shells Deflection Dynamic of optimum optimization problem columns, Maximization plates can The to the continuous Optimization according on comprehensive references t i t l e d organized exists involving Structures(1980)[18]. dynamic dimensional and elements special response can problems be divided into two follows. state dynamic a r t i c l e s chronological three structures as steady transient Published UNIT material. l i t e r a t u r e The the HYSTERETICALLY l i t e r a t u r e structures structural edition of TIP Substantial e l a s t i c v i c i n i t y v a r i a b l e - c r o s s - s e c t i o n discussed BEAM the order on o s c i l l a t i o n response forced are as steady state follows. o s c i l l a t i o n Icerman in (1969)[19] 61 formulated a constraint that the given deflection a minumum-weight of at the for bending a axial plate Plaut deflection to the of a the (1973)[22] optimal two authors The of a and response of of a design of on structures the (1970)[20] used to the broader optimality of a periodic beam, the loading, structure. potential bending both of prescribes of technique for a problem that mutual Rayleigh-Ritz of of obtained under for work truss. constraint condition design amplitude rod, bending a a virtual applicable structures point with Mroz formulated stationary optimality the the i s specified used on which design of on application. motion design p r i n c i p l e derive of problem bound load (1971)[21] minimum-weight subjected of point structures. of upper formulation conditions the an amplitude variational class sets design a He used energy beam. to static the to Plaut approximate and dynamic d e f l e c t i o n s . In with a r t i c l e s undamped by e l a s t i c damping should hot-arm vibration, resonance cantilever the free be is minimum-weight investigated. and Plaut structures. In more In effective damping design interest. of subjected with Mroz included. of beam, t i p , Icerman, a the In a constraint unit on have dealt r e a l i s t i c cases particular at the the h y s t e r e t i c a l l y to they case fundamental following damped the sandwich harmonic point the amplitude, t i p of load at is 62 The a governing h y s t e r e t i c a l l y beam 3 i s given 3 2 E 2 equation damped 3x Take t r i a l y(x,t) uniform y(x,t) 3 + 2 the transverse or vibration non-uniform of cantilever by: [I(x) 3x for 2 I(x) 3 y(x,t) 2 ] + m(x) 3x 3t u solution = y(x) - y(x,t) 3 3t 2 of = f(x,t) . the form 2 (4.2.1) cos(cot), (4.2.2) for H(-co) f ( x , t ) = P 6(x-L) cos(cot) . (4.2.3) |H(w)| For co < co H,(co), with H T ( - H(co) 1f the unit O ) ( C at O a function complex resonance C i s 2 - C co,, O 2 ) frequency = response for a i + /{(co -co ) that i s co, y ( x ) , I = Substituting 2 2 »>co 2 (4.2.4) + Uco ) } 2 an unknown I(x) by 1D0F s y s t e m . 2 |H,(w)| y c a n be a p p r o x i m a t e d i e . , = Note which 2 at present. For convenience l e t and m = m(x). (4.2.2), (4.2.3) and (4.2.4) into (4.2.1) y i e l d s : H(-co) E ( 1 + / v) [ l y " ] B - co my 2 = P 5(x-L) . (4.2.5) |H(co) | where the 4.2.1 ( )' and arguement ( )", once respectively, and twice OPTIMALITY CRITERION CANTILEVER BEAM AT THE FREE TIP with denotes respect to differentiating x. FOR T H E MINIMUM-WEIGHT EXCITED BY A UNIT HARMONIC SANDWICH POINT FORCE 63 If then y y is gives principle, E , Jn a kinematically a stiffer co < when „ Iy" dx 0 and by analogy to y(L) y Rayleigh's P (4.2.6) |H(u)| the principle of minimum potential energy Im{H(-co)} Iy" dx £ 2 y(L) P / (4.2.7) |H(u)| 0 where by of Re{H(-co)} £ 2 . fn vE and Consequently 0 j i „ my dx Sn 2 system. approximation co^ , T co - 2 admissible } Re{ and imaginary A } Jm{ part sandwich denote, of the beam respectively, taking the real has the arguement. depicted in Figure 4.3 properties: m(x) = where core the 4B{p D+p d(x)} 1 I(x) 2 p, and p , and the cover-plates respectively, 2 cover-plate. between the to bending the designs which I are The with constrained densities maintains the physical it of does a not beam. corresponding to mass elastic have the modulus Consider tip the for separation contribute deflections same of directly two y amplitude, such and y, i e . , Re{li(-co)} ReiH(-co)} y(L) the the but E (4.2.8) 2 is stiffness I, 4BD d(x) are and core cover-plates and = = P |H(w)| y(L) P |H(co)| or E/Q Iy n 2 dx co j£ my dx 2 2 = Ej|j Iy" dx 2 and Im{H{-oo) y(L) } Im{U(-co)} = P |H( )| W y(L) P |H(«)| - W 2 /Q my dx 2 (4.2.9) 64 Figure 4.3: A v a r i a b l e - c r o s s - s e c t i o n sandwich beam. or uE /Q Iy" dx = 2 * E /Q Combining /Q ( I - I ) If y of Ey" 2 - with [Ey" 2 ( p 2 - 2 / D where t J I 2 ) y kinematic y(0)=0 2 i s 2 = 2 a design with the Combining If y of G 2 ) y 2 ] and > (4.2.6) y i e l d s : 0. (4.2.11) , 2 I (4.2.12) 2 I (4.2.13) same £ constant, cannot t i p (4.2.10), y" dx design / H positive design (I-I) (4.2.8) y(L)=X that vEf^ 2 (4.2.10) constraints: (4.2.11) I ( p d x . s a t i s f i e s y'(L)=0 G 2 (4.2.9), design w l y " be s a t i s f i e s heavier it follows than any from other amplitude. (4.2.8) 0. then and • (4.2.7) y i e l d s : (4.2.14) 65 vEy" = 2 with G l , (4.2.15) kinematic y(0)=0 where y'(0)=0 G i s 2 that design with I For thus design the I same constant, cannot t i p near the 1f optimality be it heavier than (4.2.12) co then follows than from any other amplitude. smaller c r i t e r i o n co a p p r o a c h e s (4.2.16) positive much a optimality y(L)=X a (4.2.14) and constraints: and co, alone is damping c r i t e r i o n light damping, adequate. force However becomes (4.2.15) as important should be used instead. The into real E[ly"]" with governing - force and equation imaginary parts. cj 4B(p D+p d)y = 2 1 of 2 constraints: at motion The (4.2.5) real part c a n be i s divided given 0, by: (4.2.17) x=L Re{H(-u>) } EI(x)y"(x)=0 E[I(x)y"(x)]'= -P (4.2.18) |H(u)| and / the imaginary i>E[ly ]" with n force = part i s given by: 0 (4.2.19) constraints: at x=L /m{H(-u)} ?E[I(x)y"(x)]'= vEl(x)y"(x)=0 -P . (4.2.20) |H(w)| For y obtained minimum-weight (4.2.19). design I(x) The problem as approximations For w much from are smaller (4.2.12) must and satisfy (4.2.15), both quite (4.2.17) formulated i s needed to find the minimum-weight than co, and light the and complicated; hysteretic design. damping, 66 design I(x) which approximates approaches thus the near design co,, the FREQUENCY LOWER THAN (4.2.12) design. force s a t i s f i e s minimum-weight MINIMUM-WEIGHT THE damping which the equations minimum-weight l(x) approximates 4.2.2 s a t i s f i e s (4.2.15) SANDWICH CANTILEVER OF T H E UNIT HARMONIC the dimensionless y = y / L D = D/L d = d/L I = 4B/L T = w M = M/EL s = s / E P = P/EL of M the is the given by: For CJ damping, the and amplitude sandwich beam sandwich given by: D - 2 T y 2 = a important and and (4.2.19), BEAM POINT DESIGN FORCE WHEN IS MUCH 2 p 2 stress given by: L 2 / E (4.2.21) 2 in s the = cover-plates - (1 +/ » > ) E D y " , and 4BDds. than beam forms u design approximates dimensionless y " = the is smaller their 2 of which (1+/*>)M much (4.2.17) CJ quantities: x / L i s as design. = s However becomes x where (4.2.17), THE FUNDAMENTAL FREQUENCY Introducing 3 and the the 1 and which light s a t i s f i e s minimum-weight two hysteretic optimality (4.2.12) design. equations In are (4.2.22) 2 MTy = 0 (4.2.23) D»/(a +Ty ) 2 where ai i s and is given d a constant by: 2 which c a n be expressed in terms of X 67 M d = — — . (4.2.24) BD/(a +Ty ) 2 The at dimensionless x=0 "at 2 boundary y(x)=0 — ~ — x=1 y(x)=X conditions are: y'(x)=0 ~ ~ M(x)=0 Re{H{~co)} M'(x)=P . (4.2.25) |H(co) | For an found undamped a series damped beam Mroz's only solutions small 5o = M = M Note + 0 + for of of TM, + T co m u c h y, 2 a here boundary i s parameter. (4.2.27) Thus are and The different conditions. and T, small method from series found for i e . , + . . . (4.2.26) + . . . (4.2.27) smaller than into the the (4.2.22), over two co, , terms x parameter equating i s terms and applying solution T given the small. of like boundary by: „ 2 „ x D of 6 . (4.2.28) 3 Substitution the M of solved (4.2.26) perturbation D 240 terms be integrating y i e l d a, = 2 2 the terms parameter (4.2.26) T, Yo = — ~ x 2 y 2 in to kind the T used forced the + conditions — the T y , of a, solution in Substituting powers Mroz problem values Y beam like boundary (4.2.27) powers and of conditions T, (4.2.28) into integrating y i e l d : (4.2.23), over x and equating applying 68 _ M = 0 Re{H(-o)} -P (1-x) |H(u)I Ba, M, p, _ = P ( 4 x - x « - 3 ) 24 p + 120 2 ( D H ( - C J ) } _ (5x"-3x -5x+3) . 5 |H(u)| 2 (4.2.29) To specify guess H^ico) Rayleigh's ji a) = i t e r a t i v e c o r r e c t l y . fundamental 2 an For frequency quotient, computer a co, scheme tentative can be is required to design I(x), the evaluated from the i e . , EBD dy"dx 2 -z 7---Z—-. L / ! B(p D+p d)y dx 2 (4.2.30) 2 ( In ) 1 the 2 case of an undamped beam v=0, the term |H(w)| is equal needed. to unity In the corresponds the to variable = a, T 10" = = 1 1010-" which an case of beam loaded plate dimensionless P a and 5 = frequency s t a t i c a l l y . thickness d(x) T = 1 1/120 p,/p 7 could zero scheme to guess p C J 2 2 = 120 = 2 / E = = the problem Figure 4.4 presents for the following =1/2 0 to 2D = 2 the in 2.54*10""/10.3*10 280 s " 2 , not 1/10 correspond in is T=0, data typical 2B 30 of a i e . , L CJ, data: 3 and i t e r a t i v e and for P 6 ( s = 1 . 5* 1 0 " 2 / i n 2 ) 2 = in (aluminum) lb. and hot-arm, 69 Figure 4.4: cantilever (T =10"") (T =0), Optimal beam and both for cover-plate excited a forces by cantilever at the free thickness a unit beam t i p . for harmonic under static an undamped point force point force 70 4.2.3 MINIMUM-WEIGHT THE FREQUENCY FUNDAMENTAL For and D (4.2.19) 2 = = 0 where a and i s 2 y " vM" study d CANTILEVER OF THE UNIT HARMONIC BEAM POINT DESIGN FORCE WHEN IS THE FREQUENCY at in SANDWICH fundamental their resonance, dimensionless equation forms are given (4.2.15) by: (4.2.31) a \ (4.2.32) i s 2 a constant given which c a n be expressed in terms of X by: M d = — — . BDa The (4.2.33) 2 dimensionless at x=0 y(x)=0 at x=1 y(x)=X Equations those p r i n c i p l e optimality for Integrating y = — 2 ~ M the x 2 and minimum loaded and are: Shield potential for sandwich equations dimensionless t-M' ( x ) = P . (4.2.32) minimum-weight c r i t e r i o n s t a t i c a l l y applying M(x)=0 Prager of conditions y'(x)=0 (4.2.31) equations s t a t i c a l l y . boundary are (4.2.34) similar design of a (1968)[23] energy to minimum-weight in forms beam to loaded applied the derive the design of a beam. (4.2.31) boundary and (4.2.32) conditions then y i e l d : (4.2.35) D P = — (1-x) . v (4.2.36) 71 where The 2 2XD. minimum-weight same as factor ~ ~ d(x) For = a that 1 /V, weight the the resonance s t a t i c a l l y loaded excitation case is except ~ (1-x). — v BDa the of for the for a i e . , P = d(x) (4.2.37) 2 case in of the cover-plates p gPL which p D/p d 1 is 2 minimum-weight small and beam is n e g l i g i b l e , that the of the a l o n e , i e . , 2 2 ".in • 7 - r where g A is <*- 2 the gravitational typical between a line of counter the same tip amplitude. are for the uniform minimum-weight both uniform Table and to t i p where c o / s the beams. cantilever a 4.2 is are part. not The and the harmonic the the the both a comparison beam are and constrained its to f i r s t three columns of Table 4.2 the sixth column is for the forth force fundamental as and beams. respective same represents cantilever where minimum-weight unit Note part beams, beam, subjected counter acceleration. uniform-cross-section minimum-weight 38) of f i f t h Both P of frequency of its common to beams are the sinfo^t) fundamental that are at the frequencies of a free of uniform minimum-weight 72 Table 4.2: A comparison cantilever the same beams and resonant their t i p uni form uniform w uniform-cross-section counterparts with amplitude common common D I uni the minimum-weight uniform d W between minimum-weight W . mm y Io W yo 0 0 1.0 N/A 1 .000 N/A 1.000 N/A 0.5 0.25 0.844 0.375 1 .185 0.386 0.4 0.20 0.768 0.400 1 .302 0.309 0.3 0.15 0.650 0.425 1 .538 0.232 0.2 0.10 0.486 0.450 2.058 0 . 155 0.1 0.05 0.271 0.475 3.693 0.077 the cover-plate where, N/A : W 0 : p 2 : g I : y 0 x=L. the applicable 60p g 2 mass density gravitational : 0 not : of material acceleration 0.08333 8.3868*10 [P/^E] 5 A l l the A l l sandwich type cantilever shown beams, in beams are uniform Figure 4.3 fixed at x=0 and and minimum-weight, and described by free are at of equation 73 (4.2.8). The negligible damped the kind section length of 60 of Table I 0 inertia reference The y of is the length I and cover-plate moment of uniform The depth the The one beam core be units. of beams are resonant mode to tip approximation the unit-square reference depth to hot-arms. this the and t o t a l tip cross beam. The length ratio f i r s t entry The unit-square respectively, section with This Triumf uniform the of no-core beam. weight, moment amplitude l i s t e d reference inclusion of the of a boundary the sandwich core. The is maintained of with increasing core thickness. inertia respectively, of cantilever radius cross of of a beam decreases The and width 2B=1. the total weight, gyration section and of t i p through beams Let w u n ^ uniform cross section, amplitude of the beams. minimum-weight 1 2D+d=1 Table where area depth in beam sectional thickness, radius the sections cross be, > cross exterior L=60, y 2D+d(0) same A l l assumed beam. thus D, be, unit-square of 60 cross proportionately d, is are is 4.1. cantilever modifications i n i t i a l core cover-plates. beams sandwich 0 the Table corresponds modification are a subsequent are out; in beam and 0 to sandwich h y s t e r e t i c a l l y . c h a r a c t e r i s t i c 4.2 the uniform uniform the is W , 4.2 the without of of and found the of comparison for Let Let in l i g h t l y amplitude of weight and cantilever width gyration 2B=1. D as beams A are of length minimum-weight its beam L=60, has uniform-cross-section 74 counter be, at with the same respectively, the total x=0 part for Note I(x) = the that tip weight minimum-weight equation amplitude. and Let W . min cover-plate cantilever and d(0) thickness beams. (4.2.8), 4BD d(x) 2 becomes more inaccurate in comparison to the more conventional, I(x) as = 2B[(2D+d(x)} d(x) increases D=0.375 almost less is zero the beam case the of error excluded at at d=0.1 and uniform Table Equation d=0.25 and D=0.475 beams 4.2 (4.2.39) is with with was d the used beam. the i l l u s t r a t i o n , the at x=0 exceeds i e . , unity, error from beam. reference The reason, In (3.5160) d(0) D. this reference the to The For are (4.2.39) 3 percent. 0.25 of {2D-d(x)} ]/l2, relative percent. than exception in 3.5 - 3 depth of 2D+d(0) the > 1, minimum-weight where 2 = d. (4.2.40) 8 The maximum depth not have exceed D's for to the two of the the beams minimum-weight depth are of the choosen beam, uniform 2D+d(0), beam, appropiately; 2D+d, but does if this 75 i l l u s t r a t i o n Table damping and at would 4.2, for subjected the be to and uniform . beam, point force with light at hysteretic the free constraint on the tip saving are beam possible is used when the instead of a i e . , 2 16 W . W material the smaller the tip structural to e f f i c i e n t the use neutral of the axis material of to amplitude. with larger e l a s t i c modulus E give amplitude. materials smaller tip MINIMUM-WEIGHT WHITE with larger damping parameter v amplitude. DESIGN CANTILEVER WITH nearest least materials tip SANDWICH 0 located shows structural FORCE tip (4.2.41) minimize 4.3 with « bending give beams frequency, sandwich (3.5160) 0 beam harmonic material minimum-weight W a complex. shows: weight -Ei* more cantilever fundamental amplitude, W s l i g h t l y OF BEAM, NOISE A HYSTERETICALLY SUBJECTED PSD, WITH TO A DAMPED RANDOM C O N S T R A I N T ON T H E RAIN TIP DEFLECTION There structural Nigam with In are just optimization (1972)[24] weight Nigam's a and studied fatigue few references in the random i l l u s t r a t i o n , a as the the vibration structural damage on subject enviroment. optimization objective viscously of damped problem functions. uniform 76 cantilever cross beam with section and constant-thickness a tip stationary-random-process and a prescribed expected rate frequency box represents in to a general random design a Both depth the of a r t i c l e s and thin a Rao structural weight, width water This tank tower studied optimization with of wall. (1984)[25] dealt The a problem p r o b a b i l i s t i c problems. locatable. suitable In the h y s t e r e t i c a l l y beam noise S PSD, method. The d i r e c t l y 0 , damped over value is the of weight and The length, out dimensional 31.5 static beam the deflection span study is the found using the feasible constraints be with no width to on on larger and 1.85 gravity beam. regime the the is for cantilever white direct evaluated a t r i a l The iterative defined by the structure. strongback than 123 inches can of sandwich force function were design rain deflection) the due under minimum-weight random constraints in problem objective of in should inches a t i p dimensional strongback to an MS carried the v a r i a b l e - c r o s s - s e c t i o n subjected (ie. search for following v a r i a b l e - c r o s s - s e c t i o n the model acceleration. environment. cantilever the of under constraints. the thickness multiobjective References not minimized simplified ground The is are a weightless. damage stress to mean assumed with box zero is the a subjected beam parameters and is acceleration p r o b a b i l i s t i c structural subjected ground The fatigue section problem more of and optimized the PSD. mass thin-walled be in kept are that inches in depth. If small by 77 prestressing the depth be should The uniform f i r s t strongback, u t i l i z e d of sandwich the TIP Consider sandwich section force in the on of s e c t i o n , the which beam cantilever beam. be, respectively, thickness, cross of If random the cover-plate section, the uniform the rain beams, the unit sandwich W , W MS • , force Table with 4.3 mode solution for kind found Table frequency white gives the 4.1 light is PSD, MS t i p hysteretic used. ' MS of tip N 0 of deflection I, <J, ' ' 1 and cover-plate inertia of deflection beams. beams noise their and inertia 2D, uniform moment 10 tip d, and CJ 0 of the evaluated I , 0 moment weight, cantilever point be and the harmonic w i l l Let cantilever in section Let PSD uniform however, separation, fundamental uniform in total the described um' N NOISE time frequency to BEAMS This weight, to pertains WHITE and were (4.2.39). total fundamental reference cross equation the pertains beams. WITH for of s t i f f n e s s . second reference frequency. with the FORCE fundamental respectively, cross beams a inches bending cantilever RANDOM R A I N of 1.85 H Y S T E R E T I C A L L Y DAMPED UNIFORM unit-square inertia the the and design the of beams sandwich of investigations minimum-weight accordance be, A cantilever at moment TO a l l maximize cantilever D E F L E C T I O N OF SUBJECTED to following v a r i a b l e - c r o s s - s e c t i o n 4.3.1 then are subjected S over 0 , the d e f l e c t i o n s . damping case to a span of A of one the 78 Table 4.3; beams subjected w T i p . to a random d uni W deflection for rain D 0 uniform force Io N 0 0 0.250 1 .000 1 .000 1 .000 0.9 0.45 0.275 0.999 1 .054 1 .056 0.8 0.40 0.300 0.992 1.114 1 . 1 32 0.7 0.35 0.325 0.973 1 .179 1 .245 0.6 0.30 0.350 0.936 1 .249 1 .426 0.5 0.25 0.375 0.875 1 .323 1 .728 0.4 0.20 0.400 0.784 1 .400 2.278 0.3 0.15 0.425 0.657 1 .480 3.428 0.2 0.10 0.450 0.488 1 .562 6.559 0.1 0.05 0.475 0.271 1 .646 22.41 0.08333 : o : 2.8l94*l0- [E/p ]2" a 2 white noise PSD or constant 3 1 N PSD 0.50 : 0 noise N 1 S white 1.0 0 w, with cantilever I where, I sandwich : 2.9747*l0 [SoirAE2pf ] 9 PSD 79 Inspection of Table with light hysteretic with white noise the fundamental the MS tip the bending mode, the no-core shows to tip located the minimize d e f l e c t i o n , MS N, a. structural materials give and mass i e . , MS with tip density is RANDOM RAIN CONSTRAINT p give 2 DESIGN case h y s t e r e t i c a l l y THE of force the However, tip for MS the damped point on FORCE ON to vibration has the the in smallest neutral use of axis the of beam d e f l e c t i o n , inversely proportional smaller OF e l a s t i c MS t i p to parameter v modulus E WITH TIP A HYSTERETICALLY WHITE rain DAMPED SUBJECTED NOISE PSD, TO A WITH DEFLECTION minimum-weight the d e f l e c t i o n , aluminum. CANTILEVER BEAM, beam, design a with to a light a unit frequency, with parametric force of subjected fundamental amplitude, random damping larger over cantilever at larger with VARIABLE-CROSS-SECTION found. for force d e f l e c t i o n . preferred MINIMUM-WEIGHT constraint random pf. materials steel t i p is and smaller structural harmonic beams J v, the a beam e f f i c i e n t the parameters In to span, nearest least the 4.3.2 beam cantilever 3 b. subjected cantilever shows: material material MS over uniform d e f l e c t i o n . beam the for damping PSD unit-square 4.3, solution white noise was PSD, 80 S 0 , a parametric approximate general and numerical shape the of the values of i t e r a t i v e l y . beam In design deflection near the solution C(x), with the free was free 4.1 i s be was optimal Thus parameters beam with f i r s t b were that free an instead. t i p The f i r s t optimized a cantilever respect for was author; was g u e s s e d shown the a, this found near cover-plate to parameters it heavier end. variable-thickness known minimum-weight which fixed not solution section should is to the and MS stiffened approximation, given and c a as parabolic shown the shape Figure 4.5 where, C(x) A l l = - a ( x - b ) the For are a Segments the to the of a automate f i l e for beams to mass for computer task mode of of the VAST program used, solution of the kind given equation cubic beam deflection finite f i e l d s , cross parameters. (residing computer) shapes were and new elements of were used (3.4.22) equal to used joint program, input data with the compatible was w r i t t e n . and c computer VAST a b A three creating segments Twenty a, A preprocessing iteration, in beam exists c a l l e d Department beam. the minimum. these program matrix, the there i s search and square. beam, deflection repetitive each L=60 unit the element function of a Engineering the the of was u s e d f i n i t e were within MS t i p Mechanical evaluate (4.3.1) weight scheme acceptance to c . confined given which iterative in + cantilever sections for 2 A was one mode used. lengths, approximate with the 81 parabolic p r o f i l e had a of the non-uniform set at two to the Gauss inertia uniform cross sectional beam over integration the two Gauss small C(x=L). with those with the was was mass of evaluated was the average element The average section assigned moment of evaluated at was assigned do design to on cover-plates a random the which stiffened and that equal-weight improvements not C(L) the of rain to heavier the the with v i c i n i t y the of are their very small. noise v i c i n i t y the that cantilever should fixed of over complicated white are Comparison emphasize d e f l e c t i o n , in symbols improvements sandwich force are other counterparts a sandwich respectively, sections. results tip are, of justify MS in minimum-weight previous shows minimum-weight and the d e f i n i t i o n s in 4.3 uniform C(0) The However constraint cross three where d i s t r i b u t i o n s . subjected length convenience, and shows beams 4.4 respective points 4.4 and consistent For beam properties. element's and Each 4.6. element. Table cantilever Figure points non-uniform f i n i t e t i p an of the The in element. same Tables shown f i n i t e the C(x=0) as mass the beam, PSD, with one with the free be of end. 82 SECTION A-A Figure A 4.5: v a r i a b l e - c r o s s - s e c t i o n sandwich cantilever beam. Figure 4.6: Finite v a r i a b l e - c r o s s - s e c t i o n element approximation sandwich.cantilever beam. of the 83 4.4: Table sandwich with Minimum-weight cantilever white noise design of beam, subjected PSD, with a v a r i a b l e - c r o s s - s e c t i o n to a random constraint on rain force the MS t i p deflection w W b a N 'D N W 0 0.5 2.16*10" 0.3 2 . 51 * 1 0 - 0.2 1.84*10- 4.4 THE 5 5 5 EFFECT PROTOTYPE(1) So been far presented. result which placing A lumped one mass noise placed 1 .657 66.0 0.390 0.281 0.389 1 .762 3.253 70.0 0.440 0.345 0.433 1 .234 6.002 OF ADDING of at WEIGHTS TO THE TIP OF A HOT-ARM mode with were 2.037 analytical would be the be at beam Blocks 0 . 105 could cantilever white 0 . 136 comfirm weights hot-arm. 0.315 It to 0 28.8 several experiments a C(L) C(0) c analytical tip analysis attenuates is computer reassuring checked the which and of has the if one was the vibrating shown that the vibration of subjected to a have could results. easily a results The effect do one of Prototyped) addition a random of uniform rain force PSD. steel the each t i p weighing of a approximately Prototyped) 1.27 hot-arm kg with 84 coolant added from MS water the drawn 5.35 in the trend. There except that currents, could •4.5 made point force force with at the as a the guide on numbers of no has steel were decreased showed for experiment are could be no clear the above (eg., less mode blocks conclusion explanations and blocks fundamental which of a less human air t r a f f i c ) y i e l d to or the Thus a a either core Table core, V axis 0 50 , 4.2 of in 0.5 d i f f e r i n g optimal two the bending has 4.3 volume not show of the lead to comparison V 0 may be to used strongback. designs cantilever to rain of and % the deflection volume random of does with harmonic presence the minimum-weight subjected unit that replacement sandwich beams or 0.5 t i p a DESIGN frequency neutral uniform STRONGBACK cantilever vibration. in v a r i a b l e - c r o s s - s e c t i o n excitation show the beam on damping uniform PSD tip designing hysteretic REPLACEMENT subjected to the beam. Studies The supply on noise increase in Hz. fundamental nearest reference 3.25 data POSSIBLE no-core appreciable hot-arm quieter damping inclusion reference 34 difference. white effect the a of the definite studies hysteretic total Unfortunately water ON A light that no a Parametric minimal D. a of various possibly material final experimental are DISCUSSION beam at smoother have to Appendix from When frequency Hz deflections recorded inside. fundamental i n i t i a l tip flowing beams models cover-plate of with the of light beam geometries. 85 The minimum-weight subjected at the are to unit fundamental tapered near a the tip subjected to the beam span the free tip random is one the an excitation more on with r e a l i s t i c Triumf has manufactured aluminum is its 11 cover-plates are s t i f f n e s s is overly subjected to a be used at in the form Figure the 4.7. of The effective damping vibration section. a capacity fixed a force feet near the are and a at of fixed the the are that in weight end. noise are The beam PSD over heavier end. excitation as But in near in light Section 3, should be material of content the of at the the force, mass and can dual hot-arm 8-foot tip entire discussed be for feet. not a bending strongback added purpose? it vibratory further A aluminum cover-plates absorber and commonly 8 The thinner mass Most length tip vibration serves chief property. Since lumped dynamic white that length. feet. rain and model. in 3 t i p cantilever frequency important absorbers with beam cover-plates sandwich aluminum absorber mass the plates short tip near non-magnetic random the lighter excitation for strongback are wide-band construction free they cover-plate specified the that flow-induced a at with rain stiffened discussion force cantilever one of with sandwich is stiffened design and of such a point frequency and a of harmonic linearly minimum-weight the design as it to can the tip shown in increases increases the system. The in the next 86 The the crude section has above attached design gradually to its strongback both harmonic replacement characteristics tapers proposed from proposed the point for a Furthermore, tip of a random the the dynamic the rain proposed wing tip. But vibration encompasses minimum-weight at airplane towards design force an strongback designs, the the design whose a absorber. design for force white is The c h a r a c t e r i s t i c s frequency design cross strongback fundamental with has f a i r l y a unit and noise the PSD. simple construct. CDRE CDVER-PLATES ABSORBER RDDT TIP NDT TD SCALE Figure 4.7: A replacement strongback design. to 5. Triumf indicated various VIBRATION hot-arms by their methods vibrating to hot-arm reducing hot-arm that amplitudes the vibration designing most its of a vibration. f i r s t an amplitudes of a an to maximum. The original hot-arms built for heavier The by quickly are light A were are to showed a effective absorbers. out tested. random In large to for force addition, transient a makes to from mode and for the effect the total of a were of were the given 1D0F noise hot-arm vibration. is absorbers white damped Since deflection the of the comes absorbers case such since which absorbers with for than f i r s t mode and the a d i f f i c u l t . heavier reduction 87 small, comparison The that designed larger the vibration hot-arm a device vibration f i r s t of in consideration hot-arm in following, found be times are tuned as capacity can mechanism series optimal many the damped P r o t o t y p e d ) that damp where hot-arm. subjected damped is tip weights testing absorber the p r a c t i c a l hot-arm the In was important to existing the parameters system the It absorber damping structures damping a amplified contribution a An is mode, of is This attached weight are HOT-ARM v i b r a t i o n . investigated. secondary the A damped the absorber amplitudes. FOR l i g h t l y increase are vibration hot-arm are narrow-band dynamic its DAMPING absorber main PSD, S^. vibration should 88 5.1 VARIOUS The devices METHODS TO two the inside the are simple The and two [27] vitreous it is be to The avoid coating, in a in a of more are out a to onto and survey period v i s c o e l a s t i c acquisition testing is devices vibration clean The in of a do devices (1980)[26]. materials a p p l i c a t i o n . coating the heat l i f e of vacuum particles such molecular is the to release cyclotron coating within However, not undesirable the and material, would radioactive s p e c i f i c to damping ASME service unlink under of by which into are: and v i s c o e l a s t i c to and d i f f i c u l t power devices. inexpensive time predict. devices damping kept material, prime passive devices molecules intensity The for arcing. of on on hot-arm, e l e c t r i c a l magnetic passive papers the be of of an must the hand absorption find other with since r e l i a b l e . damping materials ruled modes other distributed unspecified f i e l d attenuating other the c o l l e c t i o n made vacuum the and On vibration number enamel gas use. vibration d i f f i c u l t retained tank. vibration interfere s u b - c l a s s i f i c a t i o n s had painted w i l l possibly concentrated reviewed devices cyclotron to distributed large active devices impractical Jones on VIBRATION and beginning are are HOT-ARM devices. electromagnetic A c l a s s i f i c a t i o n s devices passive f i e l d THE are: active From major ATTENUATE a chains enviroment of d i f f i c u l t to materials for available time 89 frame. S l i p considered helpful to by Triumf feature promote A if Various mass, forms of But such can devices designing a to hot-arm where hot-arm vibration hot-arm has be an place the of Prototyped) Hz) is damped mode it w i l l the other 5.2 AN operate such l i g h t l y , in the is needed. has a dynamic vibration A its the are the dynamic amplitudes hot-arm tip since the small, free tip maximum f i r s t mode. ( f i r s t absorber at tuned l i t t l e which d i f f i c u l t . is respectively with a damping. arm at is secondary mechanism is been smooth device that resonances an made damping than deflection e f f i c i e n t l y of for Hz to the the Because and 5 The a second and 15 the first interaction with modes. OPTIMAL PROTOTYPE(1) For absorber are are (1979)[29]. hot damping separated This consideration a predominantly resonances and of vibration well more larger important an designs. called designed secondary has contact and Hunt amplitudes joint absorption spring times is vibration in s t i l l , by lap hot-arm vibration many This location new surfaces reviewed magnified i n i t i a l riveted its [28], absorber amplitudes. a secondary are vibration best in the s l i d i n g absorbers makes in concentrated secondary are damping many the design to a of DYNAMIC ABSORBER FOR THE HOT-ARM years there dynamic vibratory VIBRATION has been vibration system, w i l l considerable absorbers reduce that, interest when s i g n i f i c a n t l y in added the 90 vibration response. Ormondroyd absorber, damper, mass and Den which Hartog consists attached is The c l a s s i c a l to subjected an to made design parameters when the Snowdon to beams. a l l is In subjected and Hoppe Warburton main It is to excited reference important The [36] a random which references fraction of i s given the i s an that spring the above, used above numerous with most as the damped rods main and system Jacquot (1974)[35], parameters white studies a and for noise extensively represent al damping. excitation. absorber force et of such the absorber some and Campbell the which Randall application harmonic viscous optimum systems, of and of contains mentioned optimized by for e l a s t i c Wirsching by excitation. the deterministic investigated 1DOF s y s t e m system studies (1982)[36] mass, charts showed (1973)[34], system following. a a umdamped continuous the of main (1968)[33] absorbers (1928)[30,31] harmonic (1978)[32] problem a PSD. in the small but done on the 91 dynamic vibration The basic continuous for strategy e l a s t i c vibration in continuous with absorbers. system the f i r s t system is equivalent f i r s t of designing such mode as the is as replaced lumped an absorber for Prototyped) a hot-arm follows: with parameters a 1D0F for main system vibration in the mode. optimal found absorber parameters for the 1D0F main system are constructing an a n a l y t i c a l l y . a n a l y t i c a l results are implemented in absorber. a. consider the b. 5.2.1 the test the LUMPED WHITE constructed PARAMETERS be of undamped in constructing of the attached the beam from beam vibration expressed absorber. FOR EXCITED A BY TRANSVERSELY A RANDOM RAIN VIBRATING FORCE WITH PSD shows approximated energy be 5.1 an parameters BEAM NOISE Figure and limitations absorber. CANTILEVER beam practical a transversely damped absorber for vibration the kinetic vibrating deflection vibrating in in at in the energy the the x=L. f i r s t f i r s t cantilever The f i r s t and mode the mode. mode lumped at can strain The x = free L can as: y(L,t) = (L)g, (t). (Note, the previous (5.2.1 ) definitions of symbols are valid unless 92 otherwise The stated.) effective which when absorber, mass, vibrating M = M,/^?(L). Similarly lumped of the t n point kinetic f i r s t beam e of It a lumped attachment energy mode. i s is as given the absorber, 2 effective given by: mass, of the cantilever by: (5.2.2) which beam r s t i f f n e s s , when gives vibrating placed the in at f £ » for the point K e same the strain f i r s t mode. the beam of energy It i s a as given the by: (5.2.3) 2 viscous i s attachment = M,^ /!// ( L ) . e f f The same the f ° the effective spring, cantilever K in the at the beam e f £ e placed gives f f M damping ratio, 7 j£f e of the beam i s 93 7 * 6/2* eff where 6 i s vibration can be lumped with the in logarithmic the f i r s t substituted spectral white given (5.2.4) mode. for c u , density, noise PSD, For in S for decrement e of response equation ^ , for 3.4.19. The random in beam the rain f i r s t force mode, i s by: 5.2.2 OPTIMAL EXCITED Having cantilever absorber lumped ABSORBER beam 'm' PARAMETERS the system denotes parameters with The of FOR A WITH as system shown main lumped Figure system the thus subscript ' m ' . i s the SYSTEM PSD parameters, modelled 5.2, and subscript 1DOF m a i n beam; NOISE c a n now b e in system 1D0F MAIN WHITE necessary and absorber system. replaced (5.2.5) BY A RANDOM F O R C E isolated discrete subscript be e effective = S o U T l M ] * , )/*?(L) • 2DOF for ^y ff c a l c u l a t i o n , the vibration the given where ' a ' the subscript the as a the denotes effective ' e f f can 94 Introducing the non-dimensional mass ratio "I - V co = m 2 forced K frequency tuning absorber = n parameters: M /M a m a M /M m' m ratio ratio f damping 7 = r co = system damping ^ J 3 white The non-dimensional given R U , f noise ; 7 m C m /2M co o = = = a a C / 2 M co m m m PSD = MS d i s p l a c e m e n t "A> /co a m a main = S„ m of the main mass, R, is by: ,7 ,7 ) a m - (1/2TT) / Z a H(r)H(-r) dr (5.3.1) MAIN SYSTEM ABSORBER SYSTEM Figure 5.2: A dynamic vibration 1D0F main absorber. system with an attached damped 95 where of H(r) the is main dimensional the mass MS dimensionless complex displacement, given displacement of the frequency in main response Appendix mass, N, E. is The given by: N = 2TTS <U R / K . For for the 3R/37 = a These be the Q m given t two with of the u nonlinear , 7 following the optimality p a r t i a l conditions derivatives: mass, Table 5.1 main ratio which shows system S . M increasing The a. (5.3.2) equations, simultaneously PSD, and 3R/3f = 0 . frequency undamped noise are = 0 main p . values absorber solved ratio, R (5.3.2) 2 mm and are the given to in obtain Appendix the dimensionless optimal excited response by R Q absorber a p t random have optimal MS respectively, E, 7 to damping displacement a - D p r t ^opt parameters force decreases with of a for n d an white monotonically 96 Table main 5.1; Optimal system absorber excited M f by a opt parameters random 7 for force a'opt with R 0.9926 0.0498 9.988 0.10 0.9315 0.1525 3.126 0.20 0.8740 0.2087 2.189 0.30 0.8249 0.2479 1 .772 0.40 0.7825 0.2782 1 .524 0.50 0.7454 0.3028 1 .354 0.60 0.7126 0.3234 1 .229 0.70 0.6835 0.3410 1 . 132 0.80 0.6573 0.3564 1 .054 0.90 0.6338 0.3699 0.990 1 .00 0.6124 0.3819 0.935 DAMPED V I B R A T I O N From only this were previous parameter damped small the which vibration weight section, built in section can absorber weighs on a the absorbers tested is apparent chosen to a white noise FOR THE P R O T O T Y P E p ) it be comparison damped and ABSORBER undamped of 1D0F PSD. opt 0.01 5.2.3 an that freely. mere weight n The pound, of a Prototyped) hot sizes is the existing 0.7 various HOT-ARM of a arm. n hot-arm very In that are 97 reported. absorber Figure constructed cantilever The spring absorber a steel a the basic tested. It design of contains a s l i d i n g - f r i c t i o n and spring mechanism plate. s l i d i n g - f r i c t i o n shows and and weight s l i d i n g - f r i c t i o n on 5.3 was The damping were a damping made ceramic of mechanism is also of damped weight, a mechanism. steel button effectiveness a and the which s l i d employing discussed in a the following. Figure 5.3: vibration Simplified absorber drawing attached to the of the hot-arm f r i c t i o n t i p . damped 98 5.2.3.1 Considerations in Selecting the Absorber Damping Mechanism In the parameters with a analytical the dashpot. My + a discussion = + the Ky plus is damping damping cycle no c s l i d i n g or the the is to equal and the excited My + = no the 1D0F by P damping system a The and and with f r i c t i o n sign is of the other f r i c t i o n with is amount equivalent energy When is but displacement P damping is per the solution viscous work viscous smaller than When cycle with per damping of coordinate system force: if of dissipated. dissipation harmonic velocity equivalent of f(y) approximately method an force, the nonlinear usual discrete sincot. the 4F/7TCJY. energy to of The same The a mechanism. advantages motion damping motion. F constructed force: the damping sincot. gives dashpot Ky sign on f r i c t i o n Consider cy harmonic minus occurs than modelled sincut, Y proportional + P = which of bigger is y the parameter a small, sinosoidal, replaces by F r i c t i o n is the absorber was actual damping discrete dependent coordinate. on the s l i d i n g - f r i c t i o n 1D0F = optimal damping. a excited ±F, the absorber f(y) where However is of the mechansism s l i d i n g - f r i c t i o n Consider damping damping a disadvantages of for uses following kinds absorber viscous absorber study P F is motion [37], a viscous 99 With a cycle true of motion velocity system since the more the viscous force large. slight the viscous damping type in the absorber [26] of radiation w i l l well out-gassing. The absorber mass sloshing is good the viscous about the container simple damping has some velocity the damping but to in velocity in the cannot is below with a the cyclotron has not materials, a viscous heat and vacuum and been found. for coating constrained-layer-damping contaminate d i s s i p a t i o n is to spring, alternative vacuum-sealing Another as the quickly resistant for absorber increase damping preferred v i s c o e l a s t i c or co-ordinate coordinate. is is system coordinate adjust the vibratory damped to viscous velocity which as a when per of increase displacement damping spring the f r i c t i o n construction Energy-dissipating damped proportional force square corresponding non-contaminating simple the corresponding the dissipation displacement a is the of to a the example, mechanism radiation, of the viscous increase A yet For than which With energy viscously without dissipated, optimal A stable large the proportional magnitude co-ordinate. grow is grow energy dashpot, co-ordinate. is cannot viscous can deteriorate the cyclotron damping produced in a f l u i d - f i l l e d if is the in vacuum by by the container complexity of justifed. mechanism relationship produced to by whose the the square impacting energy of of the the 100 absorber impact tip mass damping barely absorber spring main as In the It excited by energy-dissipating and multi-layer a at simple construction The depends adjustment vibration and of the levels f r i c t i o n vibration the 7_ damped care unit. The to i s by absorber, for R Warburton the small a riveted There mechanism the the from be to 5.4 R R deviations was which and is yet Q the shows p t damping for and adjusted operating (1982)[40]. response are construction must sensitive in of spring found. in Figure a the absorber s l i d i n g - f r i c t i o n unit for include mechanism been taken similar increase not damper investigation damping the amplitudes. reported appreciably has damping percent A a [39]. spring. The no becomes force the the has absorber mechanisms damping. of on it was of Lanchester harmonic s l i d i n g - f r i c t i o n r e l i a b i l i t y mechanism since the weight damped of hot-arm spring absorber j o i n t - s l i p p i n g point. to of viscously a cantilever superior in that damping on this the the with reduce only), root-slipping variations to absorber when steady absorber discontinued the a An absorber balance known energy-dissipating endless to motion i s possible the limit as [38]. a b i l i t y when horizontal damper. Other Its enough efficient mass mass tested. strong ( i . e . not main increases mass. Lanchester the was vibration made is on condition magnitudes the a at contours range For of a of f dashpot does not increase from the optimal 101 Figure mass . 7 5.4: upon Dependence absorber within a of the parameters, contour the MS f response displacement and 7 . For exceeds of the values the of main f and optimal MS a, displacement values, f-f by less 7 A - 7 A . than O P T , stated *=0- ' 1 percentage. 7 -0.01. M '+' Ref.[40] optimal 102 absorber parameters. allowable errors 5.2.3.2 The for M K 7 m lumped m Damped Vibration = M,/WU = 1/(2.9457) = 2 of the f i r s t model, damping the Absorbers on a are 0.115 P r o t o t y p e d ) mode, given l b * s 2 evaluated hot-arm from i t s parameters are solving two by: / i n = M,co /V/ (L) 2 = 1(32) /(2.9457) * £/2TT <* 0.003. 2 this small not much different nonlinear f of the For to f r i c t i o n smaller. parameters in element 2 m for Hot-arm vibration f i n i t e be Testing Prototype(1) The may But 1.0) opt a n , 7 no 116 l b / i n the optimal from the equations shows = 2 of absorber case Appendix difference 7 E for m = for three 0; a range of decimal the u (0.01 places in give an Vopf d Table 5.2 and the experiments account of each u, the the pressure s t a i n l e s s absorber adjusted was made as was force steel was the small of the as with f i r s t between plate u n t i l accompanying the four tuned the values to f Q p ceramic f r i c t i o n RMS d e f l e c t i o n possible. figures damping of the of t n. and button For then and mechanism hot-arm t i p 103 Table were 5.2: Test attached to M Fig. l b * s No. results the the damped Prototyped) f M a 2 for absorbers hot-arm t opt 7 which tip. a N 4. R opt / i n in 5.5 0.00 5. 6a 0.57*10- 2 0.05 0.965 free 5.6b 0.57*10- 2 0.05 0.965 0 . 109 5.7a 1.14*10" 2 0.10 0.932 free 5.7b 1.14*10" 2 0.10 0.932 0.152 5.8a 1 .71*10" 2 0.15 0.903 free 5.8b 1.71*10" 2 0.15 0.903 0 . 183 N/A N/A N/A 2 15*10" 9 N/A 14*10" 9 4.49 4 .7*10" 9 N/A 7 .0*10" 9 3 .1*10- 9 3 .9*10" 9 5.25 3.14 N/A noi se 2.26 where, 7 : 3 the absorber are R opt mass N : not : damping the t * i e from the with an free : from the analytical one but attached MS peak two are the the desired analytical optimal study; they values. dimensionless alone the values ratio measured measured 5Hz), system numerical tip in MS displacement of the main study. deflection the peaks for spectrum near u m for vibration at CJ^ the for main near the main system absorber. f r i c t i o n damping mechanism is disengaged but 104 it takes no account of the structural damping in the from the absorber. noise : the vibration N/A : small measuring not 5.5 deflection hot-arm when panel. It Figure spectrum of accounting different two and f, < in the f for In 2 f the hot-arm the proportion peaks 7 t i p enviroment. the in hot-arm and to in the absorber to the mass. co of the the RMS the and Let peaks, mass moved hot-arm and Both motion was of They two f r i c t i o n motion (not 5Hz). d i r e c t i o n s . When of t i p . (= m the r e l a t i v e the absorbers hot-arm mode 2 CJ^ undamped dissipation relative at absorber opposite energy this f through when below and hot-arm show the one the 5.8a frequencies contributed of peak damping) to and centre moved rate and deflection above of flowing prominent attached mode and was structural undamped attenuation 2 test f, between damping was increased in the two peaks attenuated. With f 5.7a direction modes added, were f, mass and 2 originating spectrum coolant 5.6a, one the same absorber and RMS a were be 2 the the jz's the shows the peaks f . equipment shows hot-arm. show noise applicable. Figure tip background as mode of shown was could the in peaks Figures s t r i c t l y be absorbers, and 5.6a, relative increased amplification 5.7a motion, further M increased and 5.8a. with before of gave the f 2 Since the increased u, causing over 105 0 5 10 15 20 FREQUENCY <HZ> Figure 5.5: Prototyped) RMS spectrum hot-arm without of an the tip absorber. deflection for the 1C6 0 5 10 15 FREQUENCY Figure 5.6a: P r o t o t y p e d ) the RMS hot-arm spectrum with an of the undamped 20 <HZ> t i p d e f l e c t i o n absorber for the attached to tip.(M=0.05) FREQUENCY Figure 5.6b: RMS P r o t o t y p e d ) attached to spectrum hot-arm the of with tip.(M=0.05) the an t i p <HZ> d e f l e c t i o n adjusted damped for the absorber 107 FREQUENCY Figure Prototyped) the RMS 5.7a: hot-arm spectrum with an of the undamped <HZ> t i p deflection absorber for attached the to tip.(u=0.10) FREQUENCY Figure 5.7b: RMS s p e c t r u m Prototyped) attached to hot-arm the of with tip.(M=0.l0) the an t i p <HZ) deflection adjusted damped for the absorber 108 FREQUENCY CHZ) Figure 5.8a: RMS P r o t o t y p e d ) the spectrum hot-arm of with the an t i p d e f l e c t i o n undamped absorber for attached the to tip.(M=0.15) FREQUENCY <HZ> Figure 5.8b: P r o t o t y p e d ) attached to RMS spectrum hot-arm the t i p . with (/z=0.1 5) of an the tip adjusted d e f l e c t i o n damped for the absorber 109 damping in the in the f r i c t i o n d i s s i p a t i o n effect the of of mostly the entire 5.7b 5.2. and For background noise measuring equipment vibration was Over addition the form the steady should transient of the of a a state shorten vibrations. _ 2 of energy The overall increasing MS , s l i d i n g as shown tip 5.8b the enviroment; by deflections Figure from was 7 shows vibration the hot-arm s i g n i f i c a n t l y . vibration was attenuated amount of secondary absorber. In addition random the the * 10 test hot-arm damped and rate peaks originating and small the stop system. both 1.71 attenuated a l l , the of = would optimally 5.8b n reduce vibratory and n damping and attenuation 5.6b, Table Over mechanism increasing further Figures in absorber. time vibration, needed a to damping by in to reducing damped absorber attenuate large 6. TURBULENCE The it can G E N E R A T I O N IN concensus be of irregular shape the turbulence the in a in roll-bond P r o t o t y p e d ) did hot-arm used. from a roll-bond with coolant A flow the not y i e l d channel from trace the same it was soldered form the one-third-length to time, the hot-arm and deflection mode at two panels the of looping coolant the several were hot-arm flow rates roll-bond of the on pattern to a as pipe the panel. the water let through. were 110 N, for vibration recorded in the hot-arm was cut size but pipe was as bent roll-bond onto t i p , of panel i n s t a l l e d was section of copper-plated pipe-type brief answers. flatten in these generation same The The A cross types constructed. flow avoid levels s l i g h t l y figure plate was a the and 6.2 would of one. conclusive panel and subject nebulous different Another made The vibration two wall turbulence any path, coolant-channel coolant-channel in panel to that flow c h a r a c t e r i s t i c s . one-third-length panel. a FLOW is channel the panel the when of is contribution of the panel. generating the in the 6.1) hot-arm compared were to a panel panels shown water shape study Figure on WATER excitation bends roll-bond of irregular The in coolant determine the roughness a turbulence to by in designer potential study flow, HOT-ARM COOLANT turbulent-flow by (shown section prudent the aggravated branching cross on THE aluminum One at a Prototyped) in Table The the 6.1. MS f i r s t •0.3' Figure 6.1: roll-bond Cross hot-arm sectional view of the coolant channel in a coolant channel in a panel. CD •0.3' Figure 6.2: pipe-type Cross hot-arm sectional panel. view of the 1 12 Table 6.1: roll-bond type of A comparison panel and of pipe-type panel water hot-arm vibration for the panel. flow rate N l i t r e s / m i n i n pipe-type 16 26*10" 9 pipe-type 35 43*10" 9 roll-bond 18 10*10 roll-bond 36 35*10" From Table panel is better. It d i f f i c u l t is differentiate 6.1 two it The to make very test basis d i f f i c u l t design is which not levels specialized it was to vibration accurate similar f a c i l i t y minimizes of possible and especially hot-arm the determine are vibration vibratory determine to very - 9 9 which s i m i l a r . measurement systems. without optimal vibration. 2 Without to a analytical coolant flow CONCLUSIONS The a wide-band simple random rain analytical excitation in discussion force model a Triumf on the of with the white noise actual hot-arm as PSD is flow-induced shown by the boundary-layer-turbulence wall-pressure. The analyses that beam bending a uniform material has v i b r a t i o n . their with nearest minimal Thus to the contribution sandwich u t i l i z a t i o n cantilever in beams of the beam a uniform beam model neutral axis minimizing are most material show the tip e f f i c i e n t to of minimize in tip v i b r a t i o n . The analyses that when section to no-core with the only fit inside unit vibration for constraint frequency white PSD, The minimum-weight with light type excitation and t i p amplitude, tapers with linearly to minimum-weight light gives the harmonic the of a in with zero design hysteretic the of a damping, 113 model the beam the smallest force point tip at the force with mode. cantilever subjected with damped fundamental sandwich show cross l i g h t l y rain to force a whose beam standard at constraint cover-plates at beam point random harmonic frequency, one for the damping, unit fundamental is square, vibration design of is unit and hysteretic t i p The for unit beam the fundamental noise a square both cantilever the free on the thickness t i p . sandwich cantilever subjected to a beam random 1 14 rain force mean with square fundamental heavier 6. 7. The with for is cover-plates t i p one and with stiffened smaller cantilever beam vibration models, the unit harmonic frequency and The materials beam The random cantilever rain beam force cantilever models, the The a and proposed previous both point The strongback t i p but the a is be t i p . The to damped increases the e f f e c t i v e increases the damping The predominantly vibration absorber of a tuned with strongback mass f i r s t an for can at noise PSD. p case give 2 of the both excitation the fundamental noise for PSD. the white is to dual entire PSD. near the be attached purpose; t i p main and it it structure. narrow-band attenuated unit noise weight hot-arm 1DOF give of in the v c h a r a c t e r i s t i c s the the effective fundamental in serves be give described absorber mode, E parameter design of end. the design the at are excitation white lighter capacity hot-arm to force absorber which fixed the for force vibration the PSD. point made in density damping design the both white for noise designs, and at mass encompasses force dynamic with vibration rain replacement section force larger harmonic random force on modulus for point white the e l a s t i c larger beam vibration near vibration with minimum-weight harmonic to with with smaller unit rain beam materials constraint d e f l e c t i o n , larger frequency 10. the PSD, with random 9. mode, noise materials smaller 8. t i p near beam white by random a damped system. The 115 11. optimal absorber excited by heavier absorber a parameters random mass the t i p The contributions coolant irregular a those for force with white w i l l give bigger main noise system PSD. The attenuation of vibration. flow nebulous to turbulence path, shape roll-bond too are of panel to the and study. branches generation in coolant coolant-channel roughness on the cross by bends flow path, section channel wall in in are REFERENCES 1. Brackhaus, megawatts thesis, of R.W. o s c i l l a t i o n of Royal generation for the Department Canada, Gregory, Proc. "The RF power Physics Columbia, 2. K . H . , and Triumf of the control of Cyclotron", University 1.5 Ph. of D. B r i t i s h 1975. and Paidoussis, tubular M . P . , cantilevers Society, London, "Unstable conveying 1966, v o l . f l u i d " , 293, pp. 513-542. 3. Crandall, Cambridge, 4. Crandall, Mechanical S . H . , editor, Mass., S . H . Random Vibration, M.I.T. Press, 1959. and Systems, Mark, W.D., Academic Random Press, Vibration New Y o r k in and London, 1963. 5. Crandall, Press, 6. 7. 8. S . H . , New Y o r k New Y o r k , Nigam, N . C . , Introduction Press, Cambridge, Blevins, R . D . , S t a t i s t i c s , Acamedic 1979. of Structural Dynamics, 1967. Mass., to Random Vibrations, M.I.T. 1983. "Vibration Vibration, in Theory McGraw-Hill, pp. induced Van Nostrand by Turbulence", Reinhold C o . , 1984, 147-209. Corcos, Acous. 10. and London, L i n , Y . K . , P r o b a b i l i s t i c Flow-Induced 9. Developments G . M . , "Resolution S o c . Amer. Bakewell fluctuations J r . , on , 1963, of v o l H . P . , a body of 116 pressure 35(2), in pp. turbulence", 192-199. "Turbulent revolution", J . wall-pressure J . Acoust. Soc. 1 17 Am., 11. v o l . Clinch, at 43(6), J . M . , the 12. of water flow", Chen, S . S . 1972, 15. the wall smooth-walled J . Sound Wambsganss, fuel 18, by pressure pipe V i b . , M.P., M.W., rods", pp. 1969, f i e l d containing v o l . 9(3), "Parallel-flow-induced Nuclear Engrg. and Design, 253-278. "Vibration axial flow", J . of c y l i n d r i c a l Engrg. Ind., structures 1974, v o l . 96, 547-552. Warburton, G . B . , Pergamon Press, Clough, R.W. Dynamics of pp. 16. of v o l . induced '14. and Paidoussis, pp. a of 398-419. vibration 13. pp.1358-1363. "Measurements surface turbulent pp. 1968, The Dynamical 1976, and pp. 318-319 Penzien, Structures , Behaviour and J . , of pp. Structures, 15-19. "Random McGraw-Hill, Vibrations", New York, 1975, 389-515. Davenport, largest gust A . G . , value of loading", "Note on a random Proc. Inst. the d i s t r i b u t i o n function Civ. with E n g . , of application 1964, v o l . 28, the to pp. 187-196. 17. Thomson, W.T., P r e n t i c e - H a l l , 18. Haug, E . J . and Distributed & 19. Noordhoff, Icerman, dynamic Theory 2nd of e d . , Cea, Parameter 1981, Jean, "Optimal d e f l e c t i o n " , Int. with pp.340-355 editors, Structures, Netherlands, L . J . , Vibration 1981, two 1609 Applications, and pp.464-474. Optimization volumes, of Sijthoff pages. structral design for given J . Structures, 1969, Solids 118 pp. 20. 473-490. Mroz, Z . , dynamic, 1970, 21. "Optimal R.H., 24. under "Approximate R.H., optimal Applied 29, and pp.469-475. pp. Rao, 4, N.C., 31, 50, Int. to some 1973, given of Applied static "Optimal J . Journal, and Quarterly pp.535-539. design Solids in 1972, of Structures, optimization AIAA for problems", R.T., "Structural enviroment", v o l . random 10, No. 551-553. S . S . , design v o l . to 315-318. design Shield, 1968, 4, ZAMM, Quarterly solutions v o l . structures", v o l . pp. structural multi-purpose Nigam, subject design loading", 1971, Mathematics, W. vibration 25. periodic Plaut, Prager, loads", structural v o l . of 23. "Optimal Mathematics, dynamic structures 303-309. deflection 22. of harmonically-varying pp. Plaut, design "Multiobjective with process", optimization uncertain AIAA Journal, parameters 1984, in and v o l . 22, structural stochastic No. 11, pp. 1670-1678. 26. Torvik, Control, 27. P . J . , editor, 1980, ASME, Applied Mechanics Jones, D . I . G . , applications", Control, pp. P . J . 27-51. Damping Shock D i v i s i o n , Applications and Damping Torvik Vibrations AMD-vol. " V i s c o e l a s t i c materials 1980, Vibration Committee, 38. Applications editor, for for for ASME, damping Vibration AMD-vol. 38, 119 28. Brown, C . B . , Proc. ASCE, "Factors J . affecting Struct. damping D i v . , in 1968, a lap v o l . joint", 96, pp. 1197-1217. 29. 30. Hunt, J . B . , editor, "Acceleration vibration absorbers", Dynamic Mechanical Engineering Publication Ormondroyd, dynamic Den and Den vibration A9-A22, 31. J . Hartog, dampers Vibration L t d . , J . P . , absorber", or Absorbers, London, "The Trans. impact 1979. theory ASME, of v o l . the 49/50, 1928. Hartog, Vibrations, J . P . , "Two degrees 4 edn., McGraw-Hill, th. of freedom", New Mechanical York, 1956, pp. 79-120. 32. 33. 34. Randall, S . E . , "Optimum vibration J . Design, Mech. Snowdon, J . C , Systems, Wiley, Jacquot, R.G. absorber", pp. 35. Wirsching, v o l . Vibration New Hoppe, for 103, and York, Engrg. pp. P.H. under absorber", 2, D.M. absorbers 198V, and J . III, and linear pp. Shock Taylor, damped D . L . , systems", 908-913. in Damped Mechanical 1968. D . L . , Mech. "Optimal D i v . , random ASCE, 1973, vibration v o l . 99, 612-616. response 36. Halsted Campbell, random Earthquake G.W., excitation Engrg. "Minimal using Struct. structural the Dyn. , vibration 1974, v o l . parameters for 303-312. Warburton, various and G . B . , "Optimum combinations parameters", Earthquake of absorber response Engrg. Struct. and Dyn., excitation 1982, v o l . 120 10, 37. pp. Den 381-401. Hartog, "Systems c h a r a c t e r i s t i c s " , edn., 38. impact J . J . P . , McGraw-Hill, Thomas, M.D. and damper Mech. New a variable Mechanical York, Sadek, with Eng. with 1956, M.M., pp. "The 1974, non-linear Vibrations, 4 th. of the 335-379. effectiveness spring-supported Science, or a u x i l i a r y V o l . 16, mass", No. 2, pp. 109-116. 39. Hunt, editor, "Acceleration vibration absorbers", Dynamic Vibration Mechanical Engineering Publication L t d . , pp. 40. J . B . , dampers or impact Absorbers, London, 1979, 87-97. Warburton, Dynamic G . B . , Vibration Engineering "A design Isolation Publications procedure and L t d . , for absorber", Absorption, Mechanical Worthing, England, P h i l l i p s , R . S . , 1982, pp.59-69. 41. James, H.M., Theory of Nichols, N.B. Servomechanisms, and M.I.T. Press, 1964. editors, APPENDIX Reynold's number of the water A flow in the Prototype(1) hot-arm. q = flow rate of flow area of of coolant water in a P r o t o t y p e d ) hot-arm panel A = f c n = number U = mean d t p M w a tubes speed of diameter = density of = absolute 20 = = i n 2 w = 1.94 = 2.36*10- . = = The flow the in a tube s l i g h t l y flattened tube of of the water water flow 2 7 0.5 U R 7.5*10" = v for panel / s f c P u 3 water v i s c o s i t y = i n a the water q d panel average R e y n o l d ' s No. n in = = t in the R A tube s l u g / f t 5 q/7A U in d t p w = t / value M w of 40 3 l b * s / f t 2 i n / s ( U / 1 2 ) ( d / l 2 ) p = t R indicates w that / M y , = this 121 1.1*10" flow is a turbulent one. APPENDIX The cross model sectional of P r o t o t y p e d ) The P r o t o t y p e d ) vibrating = 0 of properties inch the f i n i t e and f i n i t e t i p hot-arm x = element the x is finite treated cantilever 122.75 beam Sect. No. for element beam hot-arm element at B beam inches. are given as with transversely its Sectional in the A 2 a root at x properties following I table. P in in 1 122.75 104.75 3.660 0.843 0.590*10-* 2 104.75 75.00 9.756 4.256 0.387*10-3 3 75.00 45.00 23.460 8.883 0.315*10- 4 45.00 1 1 .93 28.960 12.722 0.305*10-3 5 1 1 .93 0.00 36.029 13.570 0.297*10-3 i n in" 2 l b * s 2 / i n f t 3 where, x,, x 2 : x A : cross I : moment p : mass A lump 122.75 coordinates sectional of per two ends of a beam element area inertia unit for of cross section volume weight of 5 pounds inches to account is for also the 122 attached hot-arm to the t i p accessories. at x = APPENDIX The effect of adding which is subjected beam free lump to mass a to a harmonic uniform point cantilever force at the t i p Consider beam a subjected concentrated in a C terms y(x,t) of = to a moment the J vibrating harmonic M(a,t) normal Lg.(t)0.(x) j where transversely uniform point applied force at x=a. cantilever F(a,t) Assume and a a solution modes f 3 the generalized coordinate gj must satisfy the equation g j (t) and + oj?gj(t) mL is If a shown in given by: = If is [F(a,t)*j(a) generalized lump mass Figure F(a,t) - M ° y ( a , t ) small adequate. The one frequency is 2 * of 4.1, a w M° the = mass M° is the = + M(a,t)*J(a)]/mL of the attached force - M ° I g . <6. given to exerted the by M° on at the x=a, as beam is a one approximation mode analysis is of the fundamental is subjected by: co /{l + (M°/mL)0 (a)}. 2 If 2 the beam harmonic point analysis gives y(x,t) =* P and force the sin(<jt)<6 beam 1 attached of P mass sin(cot) at x deflection (L)<)>, (x)H, (co) , where Hi (co) 1 beam (a) modification mode beam. = mL[co -{ 1 + (M°/mL)c4 2 2 ( a ) }co 123 2 + /cco co]. 2 = L, one to a mode 124 If the deflection is given for point l i g h t l y force damped is beam Psin(co t), the n with viscous resonant parameter c by: ( L ) 0 , (x)v/{l + ( M / m L ) 0 ( a ) } P 0 , y(x) harmonic o 2 — mLccj, The in beam deflections Table The of adding viscous subjected If to a the moment = lumped to harmonic lumped cantilever a damping a the M(a,t) various other situations a uniform point rotational beam exerted at by x=a the rotational cantilever force at the stiffener and of x=a+A stiffener K ° { y ' ( a + A , t ) - y ' ( a , t ) } = stiffener as is beam free K° If the stiffener is adequate. The CJ 2 One = P mode small shown given approximation must - ( K ° / m L ) {cV, ( a + A j - c V , ( a ) « - ( K ° A 2 / m L ) 0 7 a lumped and sin(cot) the at ( a ) g frequency 2 2 2 2 of x=L, « which is attached in to figure 4.2, by: i a gives one the mode analysis equation that s a t i s f y } g, 2 . the stiffened is a damping C° is subjected to one analysis mode deflection y(x,t) a beam is given by: (a). viscous beam 1 with K°{Lg.0!(a+A)-Zg.*!(a)}. modification * £0 + ( K ° A / m L ) c 4 7 p a r a l l e l a coordinate fundamental If of is generalized gi+u?g, given t i p is i the are 4.1. effect lumped for P s i n t u t ) * , ( L ) 0 , ( x ) H , ( u ) , a added harmonic to point gives the K° in force beam 125 where Hi'(u) « mL[{co + + If (K°A /mL)0l (a)}-£j } 2 2 the 2 / {cu + (C°A /mL)0l 2 2 harmonic beam deflection y(x) = is point given 2 2 force 2 The beam Table 2 deflections 4.1. for is Psin(to t), the n resonant by: P0, ( L ) 0 , m L { c e o + ( C ° A / m L ) 01 ( a ) }co]. 2 (x) ( a ) } v / { ^ + ( K ° A / m L ) 01 2 various other 2 2 (a)} situations are l i s t e d APPENDIX Experimental weights at The study the let through arm panel were of N, at a at block No. of placing has 34 on flow Hz of the steel at fundamental t i p and blocks of of of twos. the of f At a was water was ground inlet weighing frequency, in t i p deflection between each increments the an e x i s t i n g 10 p s i vibration at litres/min blocks the for 60 of blocks MS hot-arm drop placing hot-arm. steel of t i p , recorded n of the deflection f effect Prototyped) pressure A total t i p a Prototyped) two b l o c k s were that the A combined a placed square of hot-arm experimented. outlet. t i p effect Prototyped) on D and 1.27 kg increments and the mean fundamental mode, • N in block No. 2 f n Hz N in 2 18 3.900 11.9*10- 20 3.800 12.8*10- 22 3.700 17.3*10" 24 3.600 10. 1 * 1o- 9 26 3.500 10.9*10- 11.2*10- 9 28 3.450 9.3*10" 4.250 5.5*10- 9 30 3.350 9.1 * 1 0 ~ 14 4 . 125 6.3*10" 9 32 3.300 8.9*10- 16 4.000 5.7*10- 9 34 3.250 7.2*10" 0 5.350 10.8*10- 2 5 . 100 9.2*10- 4 4.900 8.9*10- 9 6 4.700 10.8*10- 9 8 4.550 10.6*10- 10 4.375 12 9 9 126 9 9 9 9 9 9 9 9 9 APPENDIX The optimum excited by absorber a random Consider a damped absorber random force parameters found with which with which main the white 1D0F noise PSD system and main noise minimizes a white 10DF non-dimensional main H(r) force for this mass is main system an attached excited by PSD. The optimal main displacement mass a absorber are below. The for parameters damped of E = mass complex displacement y K / P •* m m = is frequency given response function by: (A+/B)/(C+/D) where A = f - r B = 2 C = ( f D = 27 rfd-r 2 7 a 2 2 f r - r 2 ) ( 1 - r 2 2 ) - M f r 2 2 7 2 2 Ul MS d i s p l a c e m e n t of the main mass is by: R U , f , 7 The 7 f r a m r ( f - r ) 2 non-dimensional given - 4 -Mr )+27 a The 2 a , 7 m ) = (1/2TT) integral was evaluated again recorded. The on UBC reference by [4] be program MTS by was that "Reduce" network. calculus of residue and to y i e l d : [41] 127 [36]. no e r r o r s evaluated called computer dr. Warburton certain integrand algebra integrated H(r)H(-r) reported to symbolic the / with the which The integral is It was have been a i d of available integrand tables a found was in 128 R U ' ' a' m f 7 7 ) n ,y ), KM,f,7 ,7 )/L(M,f = a a m m where L(M,f ,7 ,7 ) = 4[Mf7| 7 7 {l"2f + + a 2 m a m f f t (l+/i) 2 + 4f 7 fa 0 M ) + 4 f 7 a7mT J 1+f ( 1 + M ) ]+4f 7 2 + 2 2 2 = = 7 [ l - f ( 2 +M ) + f ( l + M ) ] + M f 7 KM,f,7 ,7 ) a 2 2 3 + a m + 4f7 7 [l f d M)]+4f 7 7 2 + 2 + 2 m m } + Mf m 7 m] 3 2 4f 7|(l M) 2 + 2 a The o p t i m a l c o n d i t i o n s w h i c h m i n i m i z e s t h e MS displacement o f t h e m a i n mass a r e g i v e n b y : 3R/3f = 0 = 2 + 7 (l+M) f +[37 (l+M) +M7 2 47 7 (l M)]f 2 2 + 3 2 2 m 7 = 4f (l+y)7«+8f 7 2 a +[ 4 2 7 These p a r t i a l T h e s e two with a m (1+M)7 a 3 3 m f (l+<.)-1+f ( 2 + A t ) " f " ( l + ^ ) a 2 d e r i v a t i v e s were non-linear standard computer 2 37|-2Mf 7 7 " M f 3 m a a l s o checked with equations were s o l v e d solution routine fl 7 2 "Reduce". simultaneously for f ^ opt and a"opt* For t h e s p e c i a l case of 7 m = 0 , simple s o l u t i o n s a r e as follows: f 2 + = 0 7 a = 7 2 a 2 7 a 7 ui f M + 4 7 fa (1+M) ] f + 7M47!(1+M)-(2 + ^) ] f - 2 a7 mf - 7a a a a 3R/3 5 m 7 a opt 7 R = 2 o p t (1 ^/2) /(1 M) + 2 + = M(1+3M/4)/[4(1+M)(1+/I/2)] . = (1/M)^[(1+3M/4)/(1+M)]2
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Vibration studies on TRIUMF resonators Lee, Jimmy 1986
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Title | Vibration studies on TRIUMF resonators |
Creator |
Lee, Jimmy |
Publisher | University of British Columbia |
Date Issued | 1986 |
Description | The Cyclotron Division of Triumf, Canada's national meson facility, initiated a study to design replacement RF resonators which when in place would improve the quality of the particle beam produced by the cyclotron. A hot-arm, a major structure of a resonator, experiences thermal-related structural deformation and flow-induced structural vibration which reduces the spatial stability of the particle beam. The scope of this report encompasses studies on the hot-arm vibration to determine the desirable characteristics for a replacement hot-arm design which would reduce hot-arm vibration. The improved beam stability will be beneficial for the types of experiments conducted now at Triumf. In the future, it will be beneficial for the proposed Kaon Factory; Triumf has requested funds to construct a kaon producing factory encircling the existing cyclotron. The vibrating hot-arm is modelled analytically to be a transversely vibrating cantilever beam. Investigations are carried out on the following areas: 1. the nature of the vibration excitation forces originating from the coolant water flowing in the hot-arm; 2. the effectiveness of adding a lumped mass, a rotational stiffener and damping, to the cantilever beam model, in reducing beam vibration; 3. the minimum-weight design of a lightly damped variable-cross-section sandwich cantilever beam, subjected to a unit harmonic point force at the free tip, with constraint on the tip amplitude; 4. the minimum-weight design of a lightly damped variable-cross-section sandwich cantilever beam, subjected to a random rain force with white noise power spectral density over the beam span, with constraint on the mean square tip deflection; 5. the optimal design of a dynamic vibration absorber to attenuate the hot-arm vibration and 6. the influence of the shape of the coolant-flow channel on the magnification of the flow-induced excitation in a hot-arm. A possible design for the replacement hot-arm is discussed. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0096914 |
URI | http://hdl.handle.net/2429/26305 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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