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Mechanical Vibrations

by: Consuelo Herman DDS

Mechanical Vibrations MAE 220

Consuelo Herman DDS
GPA 3.78


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This 32 page Class Notes was uploaded by Consuelo Herman DDS on Tuesday September 8, 2015. The Class Notes belongs to MAE 220 at University of California - Davis taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/187477/mae-220-university-of-california-davis in Engineering Mechanical & Aero at University of California - Davis.

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Date Created: 09/08/15
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30 Figure 614 System analyzed in Example 65 6 Solution The equations that relate the forces acting on the system are N Mg where g is the acceleration due to gravity friction F MN pkMg d2 Md i Zf sum of all forces acting on the system so that dzx dx Mdlz f kMg k1 12quot B or upon transforming into the sdomain 1 MS2 B kl k2X A kMg 693 where the applied constant force f is equal to A b Determine the value of the damping dashpot coef cient 8 for which the system is critically damped in terms of the mass M and the spring constants kl and k2 Solution The roots of the characteristic equation M52Bsk1 k20 are given by B l i 2 l2 rhrZ 2M 2M3 4ltkl kle Critically damped behavior is achieved if the radical term is39equal to zero so that B2 4kl k2M or B 2kI k2M392 39 694 c Determine the response xl of the system Solution The Laplace transform xs ofthe system response is obtained from equation 693 c sMs2 B k1 k2 cM B k k 2 lZgt 53 5 E E 3 5 srl s rz x5 where for the values given in Figure 614 c A 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tiffMM 395 A ag8amp5 13 a saga51a slaecflaw 42 137ch cldu z l 34 quotw cm W c 7 223 j 1 73 3e mj x pawn cmoa Ma MM m m a Mec mzka 7 r Amcfwzt PM W s7ww W 0 Wdz t 4 pm em 3539 my 6e jQWd Mao151 60 Fahd 150113 Md chc 7 664404516 wkm 660 mgfohh c m w b fid 6 Cdml EMU w ATPquot MAWC F W433 7 c W n 3 mam power a m a H y M Ahmad Sto a m af a W cm opwm a naw6m 901 C 5C 225774 EMquot 2 6 m m mz zec 7431 q wawsw a7 42 s ucme E b 7 at 34 antsdew 0 mjlc dyrec QZ fmzw 358 szx bat St weava 39w 4 Mpwi we 8014 0 QLM mfcdS wae a rm 74444 6cm x pr mcl mfMf 4411 42 Wm 87 spa z Jamij a 1 ow ml 25514ij 04472 ham lad0147 am Ma 74260 We me Siu Vl J Ladw mms 97 wabJe chat 544 W ha vm tfmy does W1 awec M w h pw m 471 Sec 0 20 72 2 TRANSMISSION or RANDOM VIBRATION ideally white ie Sxw So a constant for all frequencies The integral 236 then reduces to EM so I l mu 2 dw 237 In order to facilitate this calculation we tabulate this integral for certain simple forms of Hw More extensive results of this kind are given in reference 4 i Hw J Ho I dw 11Afj11 ii Him quot31 3 MA m11 Ao BaA0 A2 L B 33932 inl Bo m m MA szz iwA1 A0 BSA0 A2143 A3031quot 28032 A182 1 Hw 2 dw 77 l AIAZA AoAg 220333 szz 1208 190 w m quot414 MAS W12 x39wAl A f l W 12 dw B Ao AzAa M A309 28032 A109 28183 B A A1142 A0143 A1A2As A1A4 AoA 238 Ergodic Processes The excitation response relations so far derived in this section have all been for ensemble averages of stationary processes Similar relations also exist for temporal averages of individual excitations and responses For example suppose that a particular excitation history x t results in the response y t The temporal auto correlation f r of the output is obtained by performing a time 20 2 35 Ida4741 7754 we camowxed ac fj m a az 39uw ekw 23 3 9 pr wa47 070 ms we 34713 2 Aka z o 7 Mam wumc 1 Me lgw IDV 8Wquot Haifa 7f and MM V me 0044ka At 7twz396 m um 0 44 fa aormj cm 6e wz uc7 v maz mzawmamJ 4A 4 4 macaw We 0 Dwdot gumn far wm m a39vcr 54 7 AA myd 07 n 65 501 mod isola v 5 44 4 sharia oleMGM a Mfg M q JAMob WW ZSMII V 410 guii nhm Md Ma ard M wbaw ra mat 41 added 1 5 M SigJpn m4 advDam A aeca v m 3 ij so 64 1 zibdl whm 544443 Wy w md 4 510 m 744 Mrf addee mass 40me would 435 753544 Wduld m recowweaL K vmww39hj ms we 6741 Haas 03924 aw Z fw e dampay F W m 5mm L14 GMM P9quot 22 35 We 00 Mawe Oldung l 02 7quot V 7 77391 72 39 70 n JJ 1 Van 73439 1 57L 25 BK Md 744764 44 4W 7 W 3 7 4 gsimc 7 29de faao az c39 bahw 41 mosr jMMU m 3 Mam MowJtd M76qu 75574451 M4 TQ b 1 0JV 7 WM 3637 quot 39tign f M wfwrin 74 701475 M0 7 2 71quotquot a ft b iz 3 lm 6731 6M HfwarezM jV z r awki MW L 43mgafmoamoe aw 752224 W mg GIQV WJ and was 4294 ch sum quotL ovas a c acrdm V I 621 MWA MZ rtl ilc mm 4 4 9 low mhu ijwscoofc 22 de hm39lamk mkr 39 w 40 I yam SAMcl be 264 5 fad Mame 31404 Md v hk 3 dof ijz m sa zra out mooz Alro ut39ous 23 30 max Ma Maw cawu 4 5 Wm gm 1 M Md MaiLi Md Lad2 12 Quad 1164 Itch quot M791 efbnghr 741 7 4 84 42 Sef faML was m b t Md 80ij Wizauf U K m rz39x K 324 41 9 KT Mdn 7fmr of h W slim 7 16 xvwa 239 289 41V 24 Z 732 MaW432w Z W45 MZ My vvMgr M VIIJ A W low QZNM 44 Aal q mz wrgw m 2063 U2 9 7 j plyM l CC p 7 agr u Il f t 7 245M 155 4452 1 Jj3 r 1 3 v 4 22 23970 439 21 LO 7750 W41 2 5054 gt 042 25 JP WanM 1 25 05 Q V lgC39 2M 35344 401 V j W EAC 0 39 w 064 39 57 JAMjg 124 r 2 429 9 5 an f L 5 0 2 Far 103 451722 J 39 343 704 1 90 3 lm J 02374 W 34 1378 5LCC L ajm f w39c 01160an A 4 7 5 i W 32 5 30quot DR 7131 0 444 quot12Hquot 0425 Kwa f fraD f J 042 34 1 idgz mrz 1 4945 dwwv7 a Va The University of California Davis Department of Mechanical and Aeronautical Engineering MAE 220 Mechanical Vibrations Professor Kamopp Room 2012 Bainer Tel 7523606 dckamoppucdavisedu Text There is no textbook Some course note will be handed out Grading Midterm Exam 25 Final Exam 50 Assigned homework should be collected in a notebook and turned in after the Final 25 Topics I Single Degreeof Freedom Review Transient Response Sinusoidal Response Damping Measures Rayleigh s Method Phase Plane for Nonlinear Oscillations II Two Degreeof Freedom Systems Coupling Effects Normal Modes III Multiple Degreesof Freedom Normal Mode Theory Rayleigh s Principle Bond Graph Multiport Transformations Frequency Response IV Introduction to Random Vibrations Autocorrelation Spectral Density Mean Square Response Calculations Vehicle Response to Random Roadway Roughness V Vibration Isolation Parameter Optimization Active and SemiActive Systems The University of California Davis Department of Mechanical and Aeronautical Engineering Professor Karnopp Mechanical Vibrations MAE 220 Historical Perspective Vibration as Science and Art By the senses of hearing and touch it was fairly obvious from early times that Vibration and sound were related It was not immediately obvious that the sensation of sound involved the Vibration of air The design of musical instruments provided some intuitive ideas about natural frequencies modes of Vibration etc but few precise details about these effects Some dates in the development of the science of vibration 1500 Leonardo recorded observations about Vibrations For example he wrote If you beat a at board you will see dust on it come together in little heaps An extension of this idea was later used to illustrate modes of Vibration of plates 1625 Mersene experimented with the Vibrations of strings 1638 Gallileo studied the laws of proportion for the Vibration of strings 1713 Taylor made a mathematical analysis of a Vibrating string Without an equation of motion he found the frequency of the rst sinusoidal mode of Vibration Taylor applied Newton s law to a differential element of the string but then used special assumptions rather than continuing on to derive a differential equation 1727 John Bernoulli studied a string with 11 equal masses attached He found the fundamental frequency of Vibration for n 23 70ve1tones were studied experimentally at this time but Bernoulli did not nd them 1733 Daniel Bernoulli began to develop the general theory of small amplitude Vibrations 1739 Euler analyzed the simple harmonic oscillator under sinusoidal forcing 1743 Daniel Bernoulli and Euler studied the beam in detail calculating modes for various end conditions Bernoulli saw that two or more modes could be excited simultaneously 1750 Euler derived his principle of linear momentum 7 essentially what is now called Newton s law It allowed simple derivations of partial differential equations for vibration 1755 Bernoulli and Euler wrote a famous paper on the vibration of beams published by the Berlin Academy 1788 Lagrange s work included results related to linearized equations of motion for vibration studies 1887 1894 7 Rayleigh s Theory of Sound provided a classical summary of the results of work in the elds of vibration and sould Some engineering applications of Vibration Computation of natural frequencies of lumped parameter or continuous mathematical models How to avoid undesirable resonances by system modification Steady state frequency response calculations Similar to alternating current calculations Self excited vibrations Machine tool chatter aerodynamic utter etc Nonlinear vibration stickslip vibration limit cycle vibration etc Rubbing and whirling of shafts Random vibration Rough burning of rockets turbulent ow forces vehicle response to roadway unevenness What will be studied in this course The mathematical description of the motion of mechanical systems Small mostly linearized ordinary and partial differential equations Steady mostly sinusoidal motion Systems with fairly weak damping Systems with fairly well separated natural frequencies


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