MECHANICS OF MACHINES II LABORATORY EXERCISES

Transcription

1 MEHANS OF MAHNES LABORATORY EXERSES. GEAR TRAN EFFENY TEST OBJETVE: To evaluate the power loss i a gear trai for differet values of power trasmitted ad the resultig efficiecies. NTRODUTON: Gears are compact, positive-egagemet, power trasmissio elemets that determie the speed, torque, ad directio of rotatio of drive machie elemets. Gear types may be grouped ito five mai categories: Spur, Helical, Bevel, Hypoid, ad orm. Typically, shaft orietatio, efficiecy, ad speed determie which of these types should be used for a particular applicatio. Gear cotact is ormally simultaeous across the etire width of the meshig teeth, resultig i a cotiuous series of shocks. These rapid shocks result i some objectioable operatig oise ad vibratio. Moreover, tooth wear results from shock loads at high speeds. Noise ad wear ca be miimized with proper lubricatio, which reduces tooth surface cotact ad egagemet shock loads. The mai advatage of the gear is the property of self-holdig, i.e. providig the immobility of the screw with its loadig oly by the axial force F (T ) 0. The efficiecy of a gear is equal to the relatio betwee the useful (output) power ad the applied (iput) power. Gear trais are used for trasmittig power from a drivig uit to a drive uit, with a chage of speed. The output from the gear box ca have a higher or a lower speed depedig o the requiremet. ower losses i the gear box which ecloses the gear trai results from viscous frictio of lubricats, slidig frictio, losses of eergy due to vibratio ad oise etc. Therefore power supplied to the gear trai is more tha power delivered to the power absorber. This experimet demostrates a method of determiig these losses. LTERATURE REVE Evaluate how the various gear types are combied ito gear drives; ad cosider the priciple factors that affect gear drive selectio. THEORY The efficiecy η of a mechaism is determied by: out η, here ip is the power iput i the mechaism ad out is the output power. This mai depedecy i egieerig is used to evaluate the theoretical ad the real (experimetally determied) values of efficiecy of a gear. To measure power losses i a gearbox, the straight forward approach is to measure the power supplied to the motor ad the powers absorbed by the power absorber so that the loss ca be foud as the differece betwee the two. The value of losses is very much less tha either of the two values measurable ad the method described above would be very iaccurate if employed. This difficulty is overcome by feedig the power output from the gearbox back ito the iput. Fig. below illustrates the priciple. ip Figure 5: The gear trai priciple Uiversity of Nairobi: SO 900:008 ertified UON/EBE/O//7

2 Let power output from gear be O, the O TO here; T torque built ito the trai ad agular velocity of the motor shaft Losses sum of losses i the motor ad i the gear trai i.e. L (total losses) L (motor losses) + L (gear losses) T The the equivalet of power supplied M L + T O i g L i M + L g + O Effiiciecy ower iput ower output ower iput for the gear traioly Note that i is a calculated value. t is ot read off the wattmeter. Oce the system attais a steady speed, the oly power supplied from the mais is the power required to overcome losses. O η + L O g For the gearbox oly L g L L T M from wattmeter readig at the same speed AARATUS Fig. shows the layout of the apparatus. A wattmeter is used for givig a readig of the power supplied to overcome losses. A variace eables a supply of variable voltage to the D motor hece a variable speed ca be achieved. The torque is icorporated ito the system usig couplig (Fig ). The speed of the motor shaft is measured by a tachometer pressed lightly at the ed of the shaft. ROEDURE Figure 6: Experimet set up. Determiatio of losses i the motor aloe. The motor is discoected from the gear tat couplig ad set to ru at a chose speed. The power supplied to the motor to overcome losses i the widig ad bearig frictio is read off the wattmeter. Ru the motor steadily at 800 rev/mi ad record the wattmeter readig as. Repeat for higher speeds icreasig the speed i steps of 00 rev/mi. i.e. at 900, 000, to 800 rev/mi.. Determiatio of losses i both the motor ad the gear trai oect the motor to the gear trai at ad build a torque ito the gear trai at couplig, by holdig oe half of with a spaer ad applyig a momet o a steel rod fixed i the couplig, ad tighteig the two halves of the couplig together. Fig. 3 shows how to determie the torque. Uiversity of Nairobi: SO 900:008 ertified UON/EBE/O//7

3 Figure 7: Determiatio of the torque Measure usig a verier protractor. After buildig the torque, ru the system at the same speeds as above ad record the correspodig wattmeter readigs represetig L T. Take four sets of readigs usig Kg, Kgs, 3Kgs, ad 4Kgs. masses. Tabulate all the results. ANALYSS. lot the four graphs of power lost i the gear trai vs. speed o the same sheet.. lot the efficiecy vs. speed for the four torques o the same sheet. DSUSSONS AND ONLUSON Discuss ad draw coclusios o your results. Uiversity of Nairobi: SO 900:008 ertified UON/EBE/O//7

4 DEARTMENT OF ENVRONMENTAL & BOSYSTEMS ENGNEERNG FEB 34 THEORY OF MAHNES LABORATORY EXERMENT FORED VBRATON OF A SNGLE DEGREE OF FREEDOM SYSTEM OBJET: To obtai curves of amplitude of the system agaist the frequecy of excitatio for various degrees of dampig, ad to derive the udamped atural frequecy of vibratio ad dampig ratio i each case. NTRODUTON A elastic system udergoes forced vibratio whe it is subjected to a cotiuous excitig force. Usually this is brought about i machiery by ubalaced rotatig parts. This vibratio ca cause damage to machies if its frequecy of vibratio coicides with the atural frequecy of vibratio of the system. Theoretically, at this coditio, the amplitude of vibratio should icrease to ifiity. practice, this is ot true because of resistaces to the vibratio from frictio, ad strai eergy absorbed by the parts udergoig deformatio such as the sprig, etc., ad they maitai the maximum amplitude at a certai level. However, the outcome of such vibratio is to cause failure of machie parts especially by fatigue. The desiger desires to kow the resoace poit of this product so that he ca desig the ruig coditio at a poit remote from resoace. The experimet demostrates a method of determiig these parameters. AARATUS Fig. shows the apparatus used. A rigid bar is pied at oe ed ad a sprig is attached towards the other ed ad fixed o the frame. A electric motor drivig a out of balace mass is mouted o the bar. The motor has a variable speed which eables differet frequecies of excitatio to be used. A dashpot fitted where oil is used to get differet values of dampig for the system. The out of balace mass rotates at half the motor speed ad the speed of the mass is the oe used i the aalysis. A speed cotrol uit gives a readig of the motor speed. A velocity pick up mouted o the bar geerates a voltage proportioal to the liear velocity ad hece the liear displacemet of the bar the poit where the velocity pickup is mouted. The voltage is small ad is amplified by a electroic uit ad fed ito a voltmeter where readigs of amplitude of vibratio are take i volts millivolts. A RO is used to display the shape of the trace produced by the vibratio for iterest s sake. The frequecy of the excitig force (ad the resultig vibratio) ca be measured with a microscope or the RO. However, the frequecy is determied from the speed cotrol uit. The calibratio factor of the pickup uit is 44 mv per mm/s. Usig this calibratio factor, the actual values of velocity ad displacemet ca be determied if so desired. 4

5 THEORY he the excitatio forces the system to vibrate, there are iertia torques about the pivot give by There are exteral dampig torques opposig iertia ad excitig torques give by where is dampig coefficiet There is a sprig torque Ka opposig iertia ad excitig torques. The excitig torque T os wt The iteral torque All exteral torques Ι p The Ka + T os wt + + Ka T os wt () This is a liear record order differetial equatio, solved by employig the mathematics differetial equatios as follows: t has bee show (i free vibratios laboratory sheet) that p where is the atural frequecy of vibratio without dampig. Ka ν a be expressed as Ka sice, ν K a The + ν ο T + os wt. () Ka T f Ka is take as A o ad recallig that os wt Real part of e i.e Re The + ν ο + A o Re (3) Ad this is d + dt ν d + dt A 0 Re (4) t has also bee established i mathematics that a differetial equatio of this form has a solutio of the form Ae. The substitutig i (4) we get

6 0 + ν jw + A 0 Re ν jw+ + A0 Re ad A o Re ν jw + + Multiplyig by the cojugate to elimiate the complex deomiator:- 0 R( ( ν jw Aoe ) 4ν w + Expadig ad takig the real part oly (5) ν Ao ( )cos wt + si wt 0 Deomiator i (5) deoted by Ζ f a agle is defied such that os ad Si z w z. ZA ( The 0 o os + SiwtSi Z hich gives 0 A o os(wt Z Z ) A o is the deflectio produced if a torque equal i magitude to the maximum value of excitig torque were applied to the system uder static coditios. The the maximum value of 0 which is 0 o.a o Z (7) 3

7 z is called the magificatio factor β Note that the amplitudes read off the voltmeter are values of ( A o Z) at the respective settigs of the system. Settig A o the value of 0 uder statig torque idetified by 0 st 0 o 0 st + The maximum value of β occurs whe / is slightly less that uit. Differetiatig β with respect to / makes the determiatio of ν possible, if the maximum poit of β o the graph of β versus / is kow. ROEDURE ith o oil i the dashpot, switch o the RO ad check that a trace is obtaied by iitiatig small free vibratios with your had. Start the motor ad ru it at 00 rev/mi i.e. the disc rotates at 50 rpm. Switch the voltmeter o ad choose a suitable scale to give you a good order of accuracy for your readig. Make sure that the pisto does ot rub o the walls of the dashpot cylider. The voltmeter poiter is very usteady. Take your readig as the average of the bad withi which the poiter dwells most. Record the voltmeter readig ad the disc speed. Repeat the above with higher motor speeds, icreasig the motor speed i steps of 00 rev.mi. After each icremet select a suitable rage for the voltmeter to give you a reasoably accurate readig. Do ot allow full scale deflectio o the voltmeter sice this ca damage it. he the system vibrates violetly, hut the positio of resoace very carefully. f you miss it, you will ot be able to get your aalysis right. Record the resoace speed ad amplitude. After locatig this importat poit, cotiue icreasig the speed ad takig readigs up to a motor speed of 500 rpm. 4

8 Repeat the above procedure with two differet samples of oil havig differet viscosities. First, put Tellus 7 i the dashpot with the oil level about 30 mm over the pisto. Remove this oil ad repeat the experimet with vitrea 90. This gives you three values of dampig ANALYSS. lot amplitude of vibratio agaist frequecy, thus determie for each value of dampig. lot all the curves o the same graph.. lot amplitude agaist / for the three cases ad obtai values of ad for each curve beig the half power poit frequecies. NOTE: The half power poits are the positios o the curve for which power requiremet is half that at resoace. Above this level, the system is sesitive to dampig. This level is used for defiig the bad width - correspodig to the half power poits. The amplitude at this level is w of the resoace amplitude. t is importat to kow ad because ν is small. f a stroboscope is available, values of agle ca be measured. ca be plotted agaist ν. f dampig is too heavy ca oly be estimated from the coditio that π at w w ν whe, ad at this poit 0 st, i.eβ. c Derive a mathematical expressio for maximum β i terms of i terms of ad, hece determie ν for each curve. 5

9 DSUSSON AND ONLUSONS ommet i details o your results. K Exciter ick- up.r.o VOLTMETER AMLFER Damper Speed cotrol 6

10 . FREE VBRATON OF SNGLE DEGREE OF FREEDOM SYSTEM OBJETVE: To determie experimetally (i) The damped atural frequecy of vibratio (ii) The dampig ratio of a system subjected to free vibratio, ad (iii) To compare the experimetal values of udamped atural frequecy with calculated estimates for differet masses of the systems. NTRODUTON: Free vibratios are those which occur after a elastic system i equilibrium is displaced from the positio of equilibrium ad let free. The system the oscillates about the equilibrium positio with the amplitude of vibratio reducig gradually from a maximum value to zero. The suppressio of vibratio called dampig is brought about by air resistace, frictioal forces ad hysteresis losses resultig from the iteral frictio of the straied elemets such as a sprig. This experimet illustrates a method of determiig parameters used i fidig how much a system is dumped durig its vibratio ad how this is related to the frequecy of vibratio. AARATUS: The sketch below shows the elemets used i this experimet. The bar of legth L is pi joied at the pivot. A sprig of stiffess K is attached to the bar at a distace a from the pivot ad achored to the frame of the rig. A additioal mass M ca be located at a distace b from the pivot. he the bar is displaced ad let go, it vibrates; ad the electric stylus mouted at the ed of the bar moves up ad dow the teledeltos paper, which wids o a rotatig drum. The stylus is heated electrically ad burs a trace of the oscillatio o the teledeltos paper. The trace produced is used to determie the dyamic parameters required. LTERATURE REVE: This dyamic system with agular oscillatios has iertia torques which i this case are the products of the momet of iertia of the assembly about the pivot ad the agular acceleratio about the pivot i.e ertia torque of system.() The iertia torque is assumed positive i the directio of positive. The dampig torques are proportioal to the agular velocity of the system. d The velocity is or dt Uiversity of Nairobi: SO 900:008 ertified UON/EBE/O//7

11 Hece, dampig torques cost. x The dampig torque cc...() the coefficiet of viscosity This torque is opposite to the iertia torque, which is why it decelerates the system. Aother torque of this system is the sprig torque at displacemet from the equilibrium positio. aka Sprig torque ka (distace from pivot perpedicular force) The sprig torque also opposes the iertia torque. All exteral torques all iteral torques Therefore, - ka Negative sigs are used because these torques oppose iteral torques (Further aalysis of this equatio will be doe i class). Note that the effect of gravity is already couter balaced at equilibrium so it does ot appear i equatio. (4) The p + + ka 0..(5) This is a liear secod order differetial equatio solved by employig the mathematics of differetial equatios as follows. This is polyomial of the form d c d ka (6) dt dt The usual method of solvig this type of differetial equatio is to use the fact that a expressio of the rt form e has bee established as a solutio to this equatio. Substitutig i (6) c r + r + ka 0... (7) from which, c ka r + r (8) Ad r λ± p λ β... (9) c k where λ ad β a There are three possible outcomes from equatio (9) as show below fλ is greater thaβ, it has bee established that the impulse give to start vibratio dimiishes i oe oscillatio, thus the system is too stiff to vibrate. t is said to be over damped. f λ is equal to β it is at the limitig coditio where oscillatio starts takig place. This is the Uiversity of Nairobi: SO 900:008 ertified UON/EBE/O//7

12 critically damped coditio. The 4 ka Ad the critical dampig coefficiet 4ka c p...(0) this case, f λ is less thaβ, dampig is small ad the system makes several oscillatio after a disturbace before it stops vibratig. This case is what is ecoutered mostly i practice ad is the oe aalyzed for this experimet. The roots of equatio (9) are complex umbers. r λ± i β λ or r λ± i whwre β λ The Ae A geeralsolutio equatios (6)ad (7)is thesum of the two. Hece Ae λ e t ( Ae it ( λ+ i ) t + Be ( λ+ i ) t it e e ) λt λt e ad Be + Be f we defie φ such that Siφ ( λ+ i ) t ( λ+ i ) t are solutios satisfyig equatios (6) ad (7)....(0) { A( ost+ isit) + B( ost isit) } { ( A+ B) + ost+ i( A B) Sit} λt { ost + DSit}...() + D ad D beig arbitrary costats. Ad osφ the equatio () ca be expressed as + D E e + D λt Si e λt { ostsiφ+ Sitosφ} ( t+ φ ) where E is a arbitrarycostat...() Equatio () represets a vibratio amplitude, the decay beig govered by the e figure below with the iitial displacemet E t as show i t e Uiversity of Nairobi: SO 900:008 ertified UON/EBE/O//7

13 The atural damped phase frequecy of vibratio is ka d c (3) From the figure above, let the displacemet be at timet, ad at time t π t t+ λt the E e Sit + φ E e E e E e t π λ t πλ λt e λ Si The e ( ) ( t + φ ) πλ Si t Si π + φ ( t + φ )...(4) πλ...(5) This is kow as the logarithmic decremet of motio. ts importace is that 0 Hece, 0 Also e π v dampig ratio 0 v a k 3... πλ π a a + e πλ...(6)...(7) coefficiet of viscosity ritical dampig coefficiet...(8) 0 ROEDURE. Determie the sprig stiffess by hagig weights ad readig the displacemet of the sprig. eigh ad measure the bar 3. Vibrate the system without ay additioal mass ad record the oscillatio o teledeltos paper for each perso i the group. 4. Repeat (3) with masses of 00g, 400g, 600g,800g ad kg. Take traces for each 5. Use a alterative set of 5 masses if the sprig is too weak for the selectio above. Uiversity of Nairobi: SO 900:008 ertified UON/EBE/O//7

14 ANALYSS. Estimate the udamped atural frequecy for each case from the equatio of motio. alculate the damped atural frequecy for each case usig data from the traces ad rotatig drum. 3. Usig eq 7 ad 8 determie for each case. 4. From the experimetally derived parameters above, calculate the udamped atural frequecy for each case, usig the formula; wd r. ω 5. Tabulate all the results i the same table. NOTE: Results from trace are more accurate if a larger umber of traces are used. K for the sprig i determied graphically. All traces should be mouted o paper ad icluded i the report uder results. DSUSSONS AND ONLUSONS From your results discuss ad draw coclusios o the experimet, givig your opiio o the importace of this experimet i practice. f there are differeces betwee the estimated ad measured frequecies, suggest possible reasos. heck whether there is ay differece i the dampig ratio whe amplitudes are small compared with the ratio whe amplitudes are large. f these are ay differeces, what could be the reaso? REFERENES Fudametals of Vibratio Mechaical vibratios- Editra ritech Uiversity of Nairobi: SO 900:008 ertified UON/EBE/O//7

15 UNVERSTY OF NAROB DEARTMENT OF ENVRONMENTAL & BOSYSTEMS ENGNEERNG 3RD YEAR 0/0 LAB ATTENDANE REGSTER OURSE ODE:... TTLE: TME S.NO GR REG. NO SURNAMES OTHER NAMES SGN LAB ATTENDED F/50/009 F/530/009 3 F/989/009 4 F/53/009 5 F/56/009 6 F/55/009 7 F/490/009 8 F/5/009 9 F/49/009 0 F/495/009 F/489/009 F/54/009 3 F/50/009 4 F/58/007 5 F/5/009 6 F/509/009 7 F/4007/009 8 F/54/009 9 F/9937/009 0 F/000/008 F/30339/009 F/497/009 3 F/499/009 4 F/58/009 5 F/500/009 6 F/488/009 7 F/30484/009 8 F/498/009 9 F/99/ F/504/009 3 F/59/009 3 F/496/ F/5/ F/30580/00 35 F/3084/ F/007/ F/9349/ F/494/ F/3964/ F/49/009 4 F/493/009 4 F/4378/ F/5/ F/59/ F/507/ F/50/ F/503/ F/55/ F/005/ F/508/009 5 F/505/ F/50/009 ROOM NO. OF STUDENTS LETURER NSTRUTOR(S) SGN UON/EBE/O//7. Uiversity of Nairobi: SO 900:008 ertified 3/4/0