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# Vibration Analysis EM 406

RHIT

GPA 3.54

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This 19 page Class Notes was uploaded by Guido Adams on Monday October 19, 2015. The Class Notes belongs to EM 406 at Rose-Hulman Institute of Technology taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/225072/em-406-rose-hulman-institute-of-technology in Engineering Mechanics at Rose-Hulman Institute of Technology.

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Date Created: 10/19/15

Coulomb Damping Todav s Obiectives Students will be able to a Understand the difference between Coulomb and Viscous C A A damping b Identify system parameters when it has Coulomb damping lt lt lt lt lt Trivia of the day Vibrations RoseHulman Institute of Technology Mechanical Engineering SDOF concept man Concept map for the solution of deterministic linear SDOF systems 2 i39 J RT 1 u N If 2 I E xx x a nvolutiou szccm Coulomb Other Na damping I Damping Examples regal Exponential decay envelope Log detremem linear decay envelope Response specmml La 13 e viscous l Numerical methods Rotating Baas Self unbalauce motion excitation 3quot a I x Force transmitted ng x Coulomb Damping dry friction EOM Damping Force F d ykN independent of velocity SEP 2 k Mk us m FBD a Motion to the right b Motion to the left SEP SEP ll X I mg l mg kX kX i Fle 1le Find EOM using conservation of linear momentum rate N ewton s 2nd Law in the Xdirection c115 a l F a mx39z kx Fs x dz 2 d gquot si nofx Vibrations RoseHulman Institute of Technology Mechanical Engineering Coulomb Damping dry friction EOM We can write this as two EOM one for each direction m c39 kx Fd X gt 0 moving right 1 m kx Fd X gt 0 moving left 2 Or in standard from 12 x X gt 0 moving right 3 a n 12 x X gt 0 moving left 4 Where Vibrations RoseHulman Institute of Technology Mechanical Engineering Coulomb Damping dry friction Solution We can solve Eqns 3 and 4 Assume lC s xR IA1cosanlA2smanl moving right i 3 A xR I A3coswnl A4sinwnl moving left 6 x0 x0 starts moving to the left x0 0 Procedure see book for details l 2 3 Vibrations Eq 1 and 2 are valid for only 12 a cycle Solve Eq 6 for A3 and A4 using lC s Endpoint of this 12 cycle becomes the IC for the moving right phase governed by Eq 5 Solve Eq 5 for A 1 and A2 Endpoint of this 12 cycle is IC for next 12 cycle Go to step 2 RoseHulman Institute of Technology Mechanical Engineering Coulomb damping Free response l m Observations Displacement Vibrations RoseHulman Institute of Technology Mechanical Engineering Hysteretic Damping structural or solid damping Experimental results indicate that for most structural materials such as steel or aluminum the energy loss is Independent of the frequency Proportional to the amplitude squared Notes Rarely is only one form of damping Dry present fnctaon A semilog plot can provide an indication of the damping present Number of cycles Vibrations RoseHulman Institute of Technology Mechanical Engineering Damping records of automobile motion Figures taken from An Introduction to Mechanical Vibrations by Steidel E E Well functlon shock o 3 3 absorber Q Yime sec E 1 0 1 2 3 Cycles E Shock absorber action is o 3 poor and damping is 397 primarily structural Time sec 2 Cycles Vibrations RoseHulman Institute of Technology Mechanical Engineering Damping records of automobile motion Figures taken from An Introduction to Mechanical Vibrations by Steidel Dry friction or rubbing structural parts is apparent N Amplntude mm Time sec 0 1 2 3 Cycles E Shock absorbers are 0 ineffective A loose fitting Time sec 3 plunger makes the damping different in each direction Vibrations RoseHulman Institute of Technology Mechanical Engineering We will use Simulink mSimulink Library Browser Eile Edit ew elp JElzl Danni Coulomb 3 Viscous Friction A discontinuity offset at zero models coulomb friction Linear gain models viscous friction y signi quot Gain quot absii1 Uffset Emil Simulink E Commonly Used Blocks 3 Continuous y Discontinuities 3 Discrete QJ Logic and Bit Operations y Lookup Tables Q Math Operations g Model Verification y ModelWide Utilities E Ports ExSubsystems g Signal Attributes y Signal Routing y Sinks Sources E UserDefined Functions 3 E Additional Math at Discrete El Control System Toolbox j E Fuzzy Logic Toolbox Ready Vibrations Dead Zone Dynamic 74 Hit Crossing Quantizer l F Hebv M Emlock Parameters Coulomb 8 nalist Coulombic and Viscous Friction mask link models viscous friction y signx quot Gain quot absii Offset A discontinuity offset at zero models coulomb frictions Linear gain n i l aalll5l6l Coulomb friction value Offset lid Coefficient of viscous friction Gain c QK I Cancel Help RoseHulman Institute of Technology Mechanical Engineering Apply Secund DEMudelZ Eile Edit Eiew imulation Fgrmat Innis elp ll39lll lNormal ggl l To Workspace CoulombS Viscous Friction Gain Blow To 39thI o Gpace39l Ready 100 odes 2 Vibrations RoseHulman Institute of Technology Mechanical Engineering One way to get the envelope is to use the A 0 Ac calera on g a Hilbert Transform Time History J minted 4 w lt N velocity ins O 10 A f 5 0 0 5 1 1395 o 05 l 15 w Pecay Envelope Nomad 5 time s a qr 2quot U If ggo gem r a 9 5 1 E g quot00 o 1 1 5 2 6 39 39 39 time 5 O 05 39l 15 E Vibrations RoseHulman Institute of Technology Mechanical Engineering Modal Analysis of Power Harvesting Host Structure John McFarland Tracy Van Zandt Don Wang Mentor Phil Cornwell ESA WR June 25 2004 w39quot39 a Los Alamos This presentation will cover the modal testing and subsequent data analysis v five 7 Performed modal 28 N MWS N u mm imam v tan V Ier l IJ Extracted 9 modes test on the host in the 070 Hz structure WW range Animated the mode shapes using ME scope w39 39 Loa lam i Roving hammer impact test used to determine the natural modes of the structure Natural Modes Design and Improve of Host location of auxilary Power Structure structure Harvesting 63 impact EM locations 13 a E 7 reference accelerometers 1Vg 3 a Les Alamns Data collected using 8channel Spectrabook and Points 1024 RT Pro SB Software Lines 400 Freq 150 deltaT 27 ms dB LBFsz G111 if I 430888 125000 1 50000 1 75000 200000 225000 250000 275000 290000 100 200 200 ME Frequency Hz Input Power Spectrum Large softtip hammer gel 1 2000 09000 05000 03000 0 02000 I i l 0 80000 60000 1 0280 Start Point i Eljilnls p W giggling 3 Cancel Exponential Force Window f i 9 L55 Alamo5 4 Frequency response data analyzed using ME scope 742 FRF measurements Used last year s modal analysis to help find the natural frequencies Extracted 9 modes from 1 to 70 Hz D A He mm lm Htlmulemanv Mm w vvvvvvvvvvvvvv anUFFmealEXElTATlDNRESPDNSE a Sew Meawpe W3 UNIS Wm H mm Mm m mm 1 MW W13 n v 2w Irv 2w W11 lZVZEN Irv2W W15 lav 2W Irv2W m W15 NV 2W Irv2W W17 lb f 2W Irv2W W18 WVZEN Irv2W W19 lB V 2W Irv2W W1lEI WVZEN Irv2W g mm 1x 201 WayN 2 mm Wzm wayN E W1t 3 2W 2W ms 2N g um mm mm Irv 2w g W1l 5 23x 2W Irv2W i M1115 25x 2W ms 2N 3 mm 2mm mmN 3 Wm 28X 2W ms 2N W1t S 23x 2W ms 2N W12EI Dlt2 i Irv2W W12t 21 20v ms 2N Hum W122 3lgtlt 2w Irv2W W123 35x 2W ms 2N W121 33x 2W ms 2N W125 3x 2 ms 2N W125 3mm l ms 2N W127 1m 2 ms 2N W128 13x 2W ms 2N lEIEIE 5 W129 182 2W Irv2W El Mzw ms 39 n 5 an 75 H mm 125 lSEI JAN U r ZUN r1 01 Z 3 a Los Alamos ME scope used to animate mode shapes Response in vertical direction applied to all points on leg No data taken in the horizontal plane p D t Los Alamos Rigid body modes of the structure Mode 2 Mode 3 Mode 4 31 Hz 78 Hz 103 Hz swa xwgistim v3 w H Rigid body motion Rigid body motion igid body twisting along Y axis along 2 axis 11 Los Alamos Twisting and bending modes Mode 5 Mode 6 Mode 8 237 Hz 33 Hz 50 Hz 1St Bending in the Y 1St Bending in the X and 2 directions and 2 directions 1 a Los Alamos Conclusion modal analysis of host structure enables optimization of tuned auxiliary structure Performed 1 experimental modal analysis Optimize tuned auxiliary structure Determined first 9 modes of the system Comparison with last year s results Last Year I This year Mode Freq Hz Mode Shape 1 21 14 Rigid X 2 275 31 Rigid Y 3 782 782 Rigid Z 4 118 108 Rigid rotation 5 139 2nd Rigid X 6 235 237 1St Torsion 7 296 33 1St Bending Y 8 382 405 2nol Bending Y 9 46 50 1st Bending X 10 655 691 2nOI Bending X 11 702 3rd Torsion 10 Lashlames iFourier Series Example T 2 w02PiT g piecewisetlt1tquot2tlt20 piecewiset 1 t2t 2 0 plot gt0 T G 0 05 1 15 2 7 a01Tintgt0 T 1 6 t0 T ann gt2Tint gcos nw0t T 2 I gco 0 n gt2Tint gsin nw0t t n bn maO evalff6 3 0167 0202 c0s314 t 00224 c0s942 t 0101 sin942 t 000808 cos157 t 00625 sin157 t plotf26t02T 0189 sin314t ank cosk w0 t 1 sn w0 t dt T 0T 2 I g sinn w0 t dt 0 T 00505 c0s628 t 00126 c0s 126 t 000562 c0s 188 t fm gta0suman k cos kw0t bn k sin kw0t k1 m m bnk sinkw0t 0159 sin628 t 00795 sin126 t 00531 sin188 t Use this as forcing to a physical system 1 06 C I IVA I IVAV 1 2 3 1 2 3 4 5 6 7 3 TF1 squot22s100 1 s2 2 s 100 Find the magnitude and phase after substituting in 10mega magomega gtabs subs sIomega TF Isubssl TF angletfomega gtargument subs sIomega TF argumentsubss I TF Now find the time response yj gta0mag 0 suman m mag mw0 cos mw0tangletf mw0 bn m mag mw0 sin mw0tangletf mw0 m1 j J 1anm magm w0 c0sm w0t anglet m w0 m anglez mWO J39 a0Wde bnm magmw0 sinmw0t Evaluate the first 5 terms to 5 decimal places evalf y 5 5 00016667 00022428 c0s3l4l6t 000081960 c0s62832t 020473 00010275 c0s94248t 10357 000020059 cos 12566 t 040940 0000054013 c0s15708 t 021091 0069599 00020953 sin31416t 0069599 00025748 sin62832t 02 473 00046241 sin94248 t 00012604 sin12566t 000041734 sin15708t 10357 040940 021091 Plot the result using the first 6 terms plot y6 t0 2T 0011 A I 00063 1V 2 4 3 9 10 11 12 13

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